problem solving questions on hcf and lcm

Lowest Common Multiples (LCM) and Highest Common Factors (HCF) Practice Questions

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HCF And LCM

Here we will learn about HCF and LCM (the highest common factor and the lowest common multiple), including how to calculate the HCF and LCM of two or more numbers and recognise when to calculate the HCF or the LCM from worded problems.

There are also HCF and LCM worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is HCF and LCM?

The HCF and LCM are two abbreviations for the highest common factor (HCF) and the lowest common multiple (LCM).

• The HCF is the largest integer (whole number) that two or more numbers can be divided by. Other names for this include the greatest common divisor (GCD) and the greatest common factor (GCF). For example, find the HCF of 8 and 12. Let’s start by writing the factors of 8 and 12. Factors of {\bf{8}} : \ 1, \ 2, \ 4, \ 8 Factors of {\bf{12}} : \ 1, \ 2, \ 3, \ 4, \ 6, \ 12 There are several numbers that occur in both lists (1, \ 2, and 4). The highest positive integer that occurs in each list is 4, and so the highest common factor of 8 and 12 is 4 .
• The LCM is the smallest integer that is a multiple of two or more composite numbers (exists within the multiplication table of each number). Another name for this is the least common multiple. For example, find the LCM of 8 and 12. Let’s start by writing the first 12 multiples of 8 and 12. Multiples of {\bf{8}} : \ 8, \ 16, \ 24, \ 32, \ 40, \ 48, \ 56, \ 64, \ 72, \ 80, \ 88, \ 96 Multiples of {\bf{12}} : \ 12, \ 24, \ 36, \ 48, \ 60, \ 72, \ 84, \ 96, \ 108, \ 120, \ 132, \ 144 There are several values that occur in both lists (24, \ 48, \ 72, and 96). The lowest of these is 24, hence the lowest common multiple of 8 and 12 is \bf{24} .

Prime factor decomposition

To calculate the HCF or LCM of two or more numbers, we can write out a list of factors or multiples as we have above, however this approach can be very time consuming and can be complicated when dealing with factors and multiples of large numbers ( 3 digit numbers in particular).

We can therefore utilise prime factors to calculate these values.

The fundamental theorem of arithmetic states that every positive integer is either a prime number, or can be written as a product of its prime factors. Every number has a unique set of prime factors.

By presenting prime factors within a Venn diagram , we can quickly determine both the HCF and LCM of the two or more numbers in the question.

For example,

8=2\times{2}\times{2}

12=2\times{2}\times{3}

The intersection of the two circles contains the highest common factor where we multiply the values within the intersection together .

Here, the HCF of 8 and 12 is equal to 2\times{2}=4.

The union of the two circles contains the lowest common multiple where we multiply the values within both circles together . 

Here, the LCM of 8 and 12 is equal to 2\times(2\times{2})\times{3}=24.

Notice that the values for the HCF and LCM match those values previously mentioned using the alternative method.

Furthermore, as the lowest common multiple is calculated by multiplying all of the factors together within the Venn diagram, the lowest common multiple can be found by multiplying the highest common factor by the remaining prime factors.

This allows us to solve problems where we are given the HCF and LCM of two numbers and we need to determine the original two numbers.

How to calculate the highest common factor

In order to calculate the highest common factor of two or more numbers:

State the product of prime factors for each number, not in index form.

Write all the prime factors for each number into a Venn diagram.

Multiply the prime factors in the intersection to find the HCF.

Explain how to calculate the highest common factor

How to calculate the lowest common multiple

In order to calculate the lowest common multiple of two or more numbers:

Multiply each prime factor in the Venn diagram to find the LCM.

Explain how to calculate the lowest common multiple

HCF and LCM worksheet

Get your free hcf and lcm worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on   factors, multiples and primes

HCF and LCM  is part of our series of lessons to support revision on  factors, multiples and primes . You may find it helpful to start with the main  factors, multiples and primes  lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Factors, multiples and primes
• Factors and multiples
• Highest common factor
• Lowest common multiple
• Prime factors
• Prime numbers
• Factor trees

HCF and LCM examples

Example 1: hcf of two simple composite numbers.

Calculate the highest common factor of 30 and 42.

2 Write all the prime factors for each number into a Venn diagram.

3 Multiply the prime factors in the intersection to find the HCF.

HCF =2\times{3}=6.

Example 2: LCM of two simple composite numbers

Calculate the lowest common multiple of 16 and 18.

18=2\times{3}\times{3}

LCM =(2\times{2}\times{2})\times{2}\times(3\times{3})=8\times{2}\times{9}=144.

Example 3: HCF worded problem

120 \ ml of red paint and 156 \ ml of blue paint are mixed together to create a tin of purple paint. The paint is then distributed equally into sample tubes. Each tube must contain the same amount of paint that must be over 20 \ ml.

What is the maximum number of tubes that can be filled with the minimum amount of paint?

156=2\times{2}\times{3}\times{13}

HCF =2\times{2}\times{3}=12.

The total amount of paint is 120 + 156 = 276 \ ml.

Dividing 276 \ ml into 12 equal shares (the HCF), we have

276\div{12}=23.

As each tube must contain over 20 \ ml of paint, we must have 12 tubes, each containing 23 \ ml of paint.

Example 4: LCM worded problem

A plumber is fixing multiple leaking pipes. Pipe A drips water every 12 seconds. Pipe B drips water every 22 seconds. Both pipes drip at the same time.

How much time passes before they next drip at the same time? Write your answer using minutes and seconds.

22=2\times{11}

LCM =(2\times{3})\times{2}\times{11}=6\times{2}\times{11}=132

132 seconds pass. Converting this to minutes and seconds is 2 minutes and 12 seconds (60 + 60 + 12 = 132, with 60 seconds = 1 minute).

How to calculate the original values given the HCF and the LCM

In order to calculate the original values given the HCF and the LCM:

• Divide the LCM by the HCF.

Calculate the product of primes of the remainder.

Determine which prime factors match each original number.

Explain how to calculate the original values given the HCF and the LCM

Example 5: calculate the numbers, given the HCF

The highest common factor of 3 numbers is 7. The product of their remaining prime factors is 30 and each number is greater than 10. Determine the value of the three numbers.

Divide the LCM by the HCF to determine the remainder.

As we already know the remainder (30), we can move on to step 2.

Using a prime factor tree, the product of primes for 30 is

30=2\times{3}\times{5}

As each value is greater than 10, the HCF 7 must be a factor of all 3 numbers and it must be multiplied by another factor. 30 has 3 prime factors, 2, \ 3, and 5 and so the original three numbers are:

A=7\times{2}=14

B=7\times{3}=21

C=7\times{5}=35

Example 6: calculate the original numbers given the HCF and LCM

Two numbers A and B have the following number properties

• HCF (A,B) = 7
• LCM (A,B) = 2310
• A is divisible by 3
• B is an even number
• 100<A<B

Determine the values of A and B.

Using a prime factor tree, the product of primes for 330 is

330=2\times{3}\times{5}\times{11}

As A is divisible by 3, two factors of A must be 3 and 7 (the HCF).

As B is even, two factors of B must be 2 and 7 (the HCF).

Writing this up so far, we have

A=3\times{7}\times{x}

B=2\times{7}\times{y}

As 330=2\times{3}\times{5}\times{11}, we have the remaining factors of 5 and 11 to place.

As 100<A<B, both A and B are greater than 100 with A being smaller than B. The only way this is possible is by making x=5 and y=11.

This means that

A=3\times{7}\times{5}=105

B=2\times{7}\times{11}=154

The solution is A = 105 and B = 154.

Common misconceptions

• Calculating the HCF instead of the LCM (and vice versa)

A very common misconception is mixing up the highest common factor with the lowest common multiple. Factors are composite numbers that are split into smaller factors. Multiples are composite numbers that are multiplied to make larger multiples.

• Incorrect evaluation of powers

It is possible to write prime factors into a Venn diagram with their associated exponent or power. This only becomes an issue when the powers are not correctly interpreted. Take for example the numbers 12 and 18. 12=2^{2}\times{3} 18=2\times{3}^{2} Here, 2^{2}=2\times{2}=4 is correct, however the same misconception could then be continued to 3^{2}=3\times{2}=6, which is incorrect. Instead, 3^{2}=3\times{3}=9. This will have an impact on the value of the HCF and the LCM.

Practice HCF and LCM questions

1. Calculate the HCF of 54 and 60.

HCF (54,60) = 2\times{3}=6

2. Calculate the LCM of 24 and 32.

LCM (24,32) = 3\times(2\times{2}\times{2})\times(2\times{2})=3\times{8}\times{4}=96

3. Two off cut lengths of ribbon measure 1.2 \ m and 80 \ cm. Each piece of ribbon needs to be cut into the fewest number of pieces the same length. What is the length of each piece?

HCF (80,120) = 2\times{2}\times{2}\times{5}=40

4. Two runners leave the start line of a 200 \ m track on the whistle. It takes runner A \ 1 minute to run 1 lap of the track and runner B \ 1 minute and 12 seconds. What distance will runner B have travelled when they next cross the start line at the same time?

Converting both lap times to seconds, runner A takes 60 seconds, and runner B takes 72 seconds.

HCF (60,72) = 2\times{2}\times{3}=12

LCM (60,72) = 5\times{12}\times(2 \times 3)=5 \times 12 \times 6 = 360

360 seconds = 6 minutes

6\div{1.2}=5 laps

5. The highest common factor of two numbers is 35. The product of the remaining factors is 33. Both original numbers contain three digits. What is the difference between the two original numbers?

Smaller number: 35\times{3}=105

Larger number: 35\times{11}=385

6. Two numbers x and y have the following number properties,

• \text{LCM}(x,y)=96
• \text{HCF}(x,y)=8
• 2<x<y<40

What is the value of x+y?

HCF and LCM GCSE questions

1. A stables needs to divide their two fields into equal sized paddocks for some horses.

Field 1 is 240 \ m^{2}. Field 2 is 160 \ m^{2}. Each horse must have at least 42 \ m^{2}.

a) What is the minimum possible area for each paddock?

b) What is the maximum number of horses that can use these two fields?

240=2^4 \times 3 \times 5 \ or \ 240=2 \times 2 \times 2 \times 2 \times 3 \times 5

160=2^5 \times 5 \ or \ 160=2 \times 2 \times 2 \times 2 \times 2 \times 5

2+3=5 \ or \ (240+160) \div 80=5

2. Given that 6480=2^{4} \times 3^{4} \times5, simplify the ratio 10800 :6480.

Remaining factors are 5 (for 10800 ) and 3 (for 6480).

3. The lowest common multiple of x and y is 2^{3} \times 3^{2} \times 5^{2} where x is a square number such that 36<x<225.

a) Calculate the exact value of x.

b) The highest common factor of x and y is 4.

Determine the value of y. Use the Venn diagram below to help you.

x=2^2 \times 5^2 \ or \ x=(2 \times 5)^2

Learning checklist

You have now learned how to:

• Use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property of any given number.

The next lessons are

• Negative numbers

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Aptitude - Problems on H.C.F and L.C.M

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• Problems on H.C.F and L.C.M - Formulas
• Problems on H.C.F and L.C.M - General Questions

Required number = H.C.F. of (91 - 43), (183 - 91) and (183 - 43)

     = H.C.F. of 48, 92 and 140 = 4.

Clearly, the numbers are (23 x 13) and (23 x 14).

L.C.M. of 2, 4, 6, 8, 10, 12 is 120.

So, the bells will toll together after every 120 seconds(2 minutes).

N = H.C.F. of (4665 - 1305), (6905 - 4665) and (6905 - 1305)

  = H.C.F. of 3360, 2240 and 5600 = 1120.

Sum of digits in N = ( 1 + 1 + 2 + 0 ) = 4

Greatest number of 4-digits is 9999.

L.C.M. of 15, 25, 40 and 75 is 600.

On dividing 9999 by 600, the remainder is 399.

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HCF and LCM

The HCF and LCM of a given set of numbers can be calculated using different methods like the division method and the prime factorization method. HCF refers to the Highest Common Factor (HCF) of two or more given numbers and it is the largest number that divides each of the given numbers without leaving any remainder. The Least Common Multiple (LCM) of two or more numbers is the smallest of the common multiples of those numbers. It is important to learn HCF and LCM in mathematics as it helps us to solve our day-to-day problems related to grouping and sharing. Let us learn about the different methods used to find the HCF and LCM of numbers.

What is HCF and LCM?

HCF is defined as the highest common factor present in two or more given numbers. It is also termed as the ' Greatest Common Divisor ' (GCD). For example, the HCF of 24 and 36 is 12, because 12 is the largest number which can divide both the numbers completely. Similarly, the Least Common Multiple (LCM) of two or more numbers is the smallest number which is a common multiple of the given numbers. For example, let us take two numbers 8 and 16. The multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, and so on. The multiples of 16 are 16, 32, 48, 64, 80, 96, and so on. The first common value among these multiples is the Least Common Multiple (LCM) for 8 and 16, which is 16. Now, let us learn how to find the HCF and LCM of numbers .

How to Find HCF and LCM?

There are various methods that are used to find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of numbers. The most common methods are:

• Prime factorization method
• Division method

Let us discuss these methods in detail.

Finding HCF and LCM by Prime Factorization

By using the prime factorization method for finding LCM and HCF, we first need to find the prime factors of the given numbers. Then, we can calculate the values of HCF and LCM by following the process explained below.

HCF by Prime Factorization

In order to find the HCF of the given numbers by prime factorization , we find the prime factors of those numbers. After finding the factors, we find the product of the prime factors that are common to each of the given numbers. In some cases, we multiply the common prime factors that have the least or smallest power to obtain the HCF of those numbers. For example, let us find the HCF of 50 and 75 by the prime factorization method.

• The prime factors of 50 = 2 × 5 × 5 or 2 × 5 2
• The prime factors of 75 = 3 × 5 × 5 or 3 × 5 2

The common factors of 50 and 75 are 5 2 or 5 × 5. Thus, HCF of (50, 75) = 25.

LCM by Prime Factorization

To calculate the LCM of any given set of numbers using the prime factorization method, we follow the steps given below:

• Step 1: List the prime factors of the given numbers.
• Step 2: After finding the prime factors, we write them in their exponent form, and then find the product of only those prime factors that have the highest power. The product of these factors with the highest powers is the LCM of the given numbers.

Example 1: Let us find the LCM of 160 and 90 using prime factorization.

• Step 1: The prime factors of 160 = 2 × 2 × 2 × 2 × 2 × 5. After writing this in the exponent form, we get 2 5 × 5 and 90 = 2 × 3 × 3 × 5 or 2 × 3 2 × 5.
• Step 2: Now, if we pick the factors with the highest power, we get 2 5 , 3 2 , and 5. The product of all these prime factors = 2 5 × 3 2 × 5 = 1440.

Therefore, LCM of 160 and 90 = 1440.

Example 2: Find the LCM of 30 and 60 using prime factorization.

Solution: Let us find the LCM of 30 and 60 using the prime factorization method.

• Step 1: The prime factorization of 30 and 60 are: 30 = 2 × 3 × 5; and 60 = 2 × 2 × 3 × 5
• Step 2: If we write these prime factors in their exponent form it will be expressed as, 30 = 2 1 × 3 1 × 5 1 and 60 = 2 2 × 3 1 × 5 1
• Step 3 : Now, we will find the product of only those factors that have the highest powers among these. This will be, 2 2 × 3 1 × 5 1 = 4 × 3 × 5 = 60

Therefore, the LCM of 30 and 60 is 60.

Finding HCF and LCM by Division Method

There are two different ways to apply the division method to find LCM and HCF. Let us learn it one by one.

HCF by Division Method

To find the HCF by division method, follow the steps given below.

• Step 1: First, we need to divide the larger number by the smaller number and check the remainder .
• Step 2: Make the remainder of the above step as the divisor and the divisor of the above step as the dividend and perform the division again.
• Step 3: Continue the division process till the remainder is not equal to 0.
• Step 4: The last divisor will be the HCF of the given numbers.

Let us understand this method using an example.

Example: Find the HCF of 198 and 360 using the division method.

Solution: Read out the following steps and relate them with the figure given below.

• Step 1: Divide 360 by 198. The obtained remainder is 162.
• Step 2: Make 162 as the divisor and 198 as the dividend and perform the division again. Here the obtained remainder is 36.
• Step 3: Make 36 as the divisor and 162 as the dividend and perform the division again. Here the obtained remainder is 18.
• Step 4: Make 18 as the divisor and 36 as the dividend and perform the division again. Here the obtained remainder is 0.
• Step 5: The last divisor, 18, is the HCF of 360 and 198.

LCM by Division Method

To find the LCM of numbers by the division method, we divide the numbers with prime numbers and stop the division process when we get only 1 in the final row. Observe the steps given below to find the LCM of the given numbers using the division method. Let us understand the method with the help of an example.

Example: Find the LCM of 7, 8, 14, and 21.

• Step 1: Divide the numbers by the smallest prime number such that the prime number should at least divide 1 of the given numbers. Here, we will divide the numbers 7, 8, 14, 21 by the smallest prime number, i.e., 2.
• Step 2: Write the quotients of the divisible numbers right below the numbers in the next row and copy the other numbers as it is. So, the next row will be written in this way: 7, 4, 7, and 21.
• Step 3: Now, for the next division step consider the above quotients as the new dividends. Repeat the process and write the quotient below the numbers. Here, on dividing 7, 4, 7, 21 by 2, we get the quotients as 7, 2, 7, 21. [Only 4 was divisible by 2 in this step, so we copy the other three numbers as it is in the next row]
• Step 4: Repeat the steps and divide the new dividends till we get 1.
• Step 5: Multiply all the prime numbers on the left-hand side of the bar to get the LCM of the given numbers. This will be 2 × 2 × 2 × 3 × 7 = 168. Therefore, the LCM of 7, 8, 14 and 21 is 168.

Note: Divide the numbers only by prime numbers .

Therefore, the LCM of 7, 8, 14, and 21 is 168.

Let us learn more about the HCF and LCM formula in the following section.

HCF and LCM Formula

The LCM and HCF of two numbers share a relationship with each other. This is expressed in the form of a formula. The HCF and LCM formula of two numbers 'a' and 'b' is expressed as

HCF (a, b) × LCM (a, b) = a × b

In other words, the formula of HCF and LCM states that the product of any two numbers is equal to the product of their HCF and LCM. To know more about LCM and HCF relationship, visit this page which describes the relation between HCF and LCM .

The HCF and LCM formula is used in many ways. For example, if any one of the values of HCF or LCM is not known, we can easily find the other without using any of the methods given in the above sections. Let us understand this using an example.

Example: The HCF and LCM of 2 numbers is 8 and 96 respectively. If one of the two numbers is 24, find the other number.

Solution: HCF of the numbers = 8; LCM of the 2 numbers = 96; One number = 24, Second number = ?

Using the HCF and LCM formula, we can find the unknown value.

After substituting the values we get, 8 × 96 = 24 × b

So the value of 'b' = 768/24 = 32. Therefore, the other number is 32.

Note: Similarly, we can find the LCM of two numbers if we know the two numbers and the HCF using the same formula.

HCF and LCM Tricks

• If 1 is the HCF of 2 numbers, then their LCM will be their product. For example, the HCF of 2 and 3 is 1, now the LCM of 2 and 3 will be 2 × 3 = 6.
• For two coprime numbers , the HCF is always 1. For example, let us take two co-prime numbers 4 and 5, we can see that their HCF is 1 because co-prime numbers do not have any common factor other than 1.

Difference between HCF and LCM

The difference between the concept of HCF and LCM is given in the following table:

☛ Related Articles

• Properties of HCF And LCM
• Least Common Multiple Formula
• Factors and Multiples

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HCF and LCM Examples

Example 1: Find the HCF and LCM of 14 and 28 using prime factorization.

HCF of 14 and 28:

The prime factors of 14 = 2 × 7

The prime factors of 28 = 2 × 2 × 7 or 2 2 × 7

The HCF is the product of the common prime factors with the least powers. The common prime factors of 14 and 28 with the least powers are 2 and 7.

Therefore, the HCF of 14 and 28 is 2 × 7 = 14.

LCM of 14 and 28:

The prime factors of 14 = 2 × 7 = 2 1 × 7 1

The prime factors of 28 = 2 2 × 7 1

Now, we will find the product of only those factors with the highest powers. This will be 2 2 × 7 1 = 4 × 7 = 28

Example 2. Find the HCF and LCM of 126 and 162 using the division method.

Solution: First, we will find the HCF of the two numbers 126 and 162 using the given steps:

• Divide 162 by 126. The obtained remainder is 36.
• Make 36 as the divisor and 126 as the dividend and perform the division again. Here the obtained remainder is 18.
• Make 18 as the divisor and 36 as the dividend and perform the division again. Here the obtained remainder is 0.
• The last divisor,18, is the HCF of 126 and 162.

Therefore, the HCF of 126 and 162 = 18.

Let us find the LCM of 126 and 162 by division method using the following steps:

• Step 1: Divide the numbers 126 and 162 by the smallest prime number, i.e., 2.
• Step 2: Write the quotient below the numbers in the next row: 63 and 81.
• Step 3: Now for the next division step, 63 and 81 will be the dividends.
• Step 4: Think of a prime number again that divides at least one of the two numbers 63 and 81.
• Step 5: Write the quotient below the numbers 63 and 81. The next set of quotients are 21 and 27.
• Step 6: Now 21 and 27 are the new dividends.
• Step 7: Repeat the steps till we get 1 in the final row.
• Step 8: Multiply all the prime numbers on the left-hand side of the bar and get the LCM of the given numbers.

Therefore, the LCM of 126 and 162 is 1134.

Example 3: Find LCM and HCF of 510 and 92.

LCM of 510 and 92

Let us find the LCM of 510 and 92 using the prime factorization method.

The prime factors of 510 = 2 × 3 × 5 × 17 = 2 1 × 3 1 × 5 1 × 17 1

The prime factors of 92 = 2 × 2 × 23 = 2 2 × 23

Now, we will find the product of only those factors with the highest powers. This will be 2 2 × 3 1 × 5 1 × 17 1 × 23 1 = 4 × 3 × 5 × 17 × 23 = 23460

Therefore, the LCM of 510 and 92 = 23460

HCF of 510 and 92

Since we know the 2 numbers and we also know the LCM of the two numbers we can find the HCF of 510 and 92 using the formula: HCF (a, b) × LCM (a, b) = a × b

HCF × 23460 = 510 × 92

Now, we can find the value of HCF by solving this equation.

HCF = 46920/23460

Therefore, the HCF of 510 and 92 = 2

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Practice Questions on HCF and LCM

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FAQs on HCF and LCM

What is the full form of hcf and lcm.

The full form of HCF is 'Highest Common Factor' and the full form of LCM is 'Least Common Multiple' or 'Lowest Common Multiple'.

What is the Difference Between HCF and LCM?

The Least Common Multiple (LCM) of two or more numbers is the smallest number among all the common multiples of the given numbers, whereas, the HCF (Highest Common Factor) of two or more numbers is the highest number among all the common factors of the given numbers.

What is the Relation Between HCF and LCM of Two Numbers?

The relationship between the HCF and LCM of two numbers is that the product of the LCM and HCF of any two given numbers is equal to the product of the given numbers. Let us assume 'a' and 'b' are the two given numbers. The formula that shows the relationship between their LCM and HCF is: LCM (a,b) × HCF (a,b) = a × b. For example, let us take two numbers 12 and 8. Let us use the formula: LCM (12,8) × HCF (12,8) = 12 × 8. The LCM of 12 and 8 is 24; and the HCF of 12 and 8 is 4. On substituting the values in the formula we get 24 × 4 = 12 × 8. This shows: 96 = 96.

What is the HCF and LCM of numbers?

The highest common factor (HCF) of the given numbers is the largest number which divides each of the given numbers without leaving any remainder. The l east common multiple (LCM) of two or more numbers is the smallest of the common multiples of those numbers.

What is the Use of HCF and LCM?

HCF can be used in the following situations:

• When we want to divide the things into smaller sections.
• To arrange things in groups and rows.

LCM can be used in the following situations:

• An event that is repeating continuously.
• For the analysis of a situation that will occur again at the same time.

How to find the HCF and LCM in Math?

There are various methods to find the HCF and LCM of numbers. The two common ways to find the LCM and HCF of the given numbers are the prime factorization method and the division method. Both the methods are explained in detail in this page in the above sections.

How to Find the HCF and LCM of Two Numbers Using the Division Method?

To find the HCF of the given numbers by division method, we follow the given steps:

• Step 1: Divide the given numbers (larger number by the smaller number) and check the remainder.
• Step 2: Make the remainder of the above step as the divisor , and the divisor of the above step as the dividend and perform the division again.
• Step 3: Continue the division process till we get the remainder as 0.
• Step 4: The last divisor will be the HCF of the two numbers.

To find the LCM of the given numbers by division method we follow the given steps:

• Step 1: Divide the numbers by the smallest prime number.
• Step 2: Write the quotients right below the numbers in the next row.
• Step 3: Now, for the next division step consider the above quotients as the new dividends.
• Step 4: Think of a prime number again that completely divides at least one of the dividends.
• Step 5: Repeat the steps till we get 1 in the final row.
• Step 6: Multiply all the prime numbers on the left-hand side of the bar to get the LCM of the given numbers.

How to Find HCF and LCM using Prime Factorization?

In order to find the HCF of numbers using prime factorization, we use the following steps:

• First, find the prime factors of the numbers separately and note them.
• We then write them in the exponent form using their powers.
• Multiply the common prime factors that have the least or smallest power to obtain the HCF of those numbers.

In order to find the LCM of numbers using prime factorization , we use the following steps:

• List the prime factors of the given numbers.
• After finding the prime factors, we write them in their exponent form.
• Then find the product of only those prime factors that have the highest power. The product of these factors with the highest powers is the LCM of the given numbers.

A detailed explanation of both these methods, with examples, is given above on this page.

How to solve LCM and HCF Problems?

HCF and LCM problems can be solved using different methods like the prime factorization method and the division method. In the case of LCM and HCF word problems, we need to understand and look for a few keywords in the problem that lead us to a clue of whether they want us to find the LCM or the HCF.

• For example, if the problem asks for the 'smallest' quantity or the 'least' quantity, we need to find the LCM.
• Whereas, if it uses keywords like the 'highest' or the 'greatest' we need to find the HCF.

HCF AND LCM WORD PROBLEMS

Problem 1 :

A merchant has 120 liters and 180 liters of two kinds of oil. He wants to sell the oil by filling the two kinds in tins of equal volumes. Find the greatest volume of such a tin.

The given two quantities 120 and 180 can be divided by 10, 20,... exactly. That is, both the kinds of oils can be sold in tins of equal volume of 10, 20,...  liters .

But, the target of the question is, the volume of oil filled in tins must be greatest.

So, we have to find the largest number which exactly divides 120 and 180. That is the highest common factor (HCF) of (120, 180).

HCF (120, 180) = 60 liters

The 1 st  kind 120  liters is sold in 2 tins of of volume 60 liters in each tin.

The 2 nd  kind 180 liters  is sold in 3 tins of volume 60 liters in each tin.

Hence, the greatest volume of the tin is 60  liters .

Problem 2 :

Find the least number of square tiles by which the floor of a room of dimensions 16.58 m and 8.32 m can be covered completely.

We require the least number of square tiles. So, each tile must be of maximum dimension.

To get the maximum dimension of the tile, we have to find the largest number which exactly divides 16.58 and 8.32. That is the highest common factor (HCF) of (16.58, 8.32).

To convert meters into centimeters, we have to multiply by 100.

16.58  ⋅ 1 00 = 1658 cm

8.32  ⋅  100 = 832 cm

HCF (1658, 832) = 2 cm

Hence the side of the square tile is 2 cm.

Required no. of tiles :

= (Area of the floor)/(Area of a square tile)

= (1658  ⋅  832)/(2  ⋅  2)

Hence, the least number of square tiles required is 344,864.

Problem 3 :

A wine seller had three types of wine. 403 liters of 1st kind, 434 liters of 2nd kind and 465 liters of 3rd kind. Find the least possible number of casks of equal size in which different types of wine can be filled without mixing.

For the least possible number of casks of equal size, the size of each cask must be of the greatest volume.

To get the greatest volume of each cask, we have to find the largest number which exactly divides 403, 434 and 465. That is the highest common factor (HCF) of (403, 434, 465).

HCF (403, 434, 465) = 31 liters

Each cask must be of the volume 31 liters.

Required number casks :

= 403/31 + 434/31 + 465/31

= 13 + 14 + 15

Hence, the least possible number of casks of equal size required is 42.

Problem 4 :

Six bells commence tolling together and toll at intervals of 2, 4, 6, 8 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together ? (excluding the one at start)

For example, let the two bells toll after every 3 and 4 seconds respectively.

Then the first bell tolls after every 3, 6, 9, 12 seconds...

Like this, the second bell tolls after every 4, 8, 12 seconds...

So, if the two bell toll together now, again they will toll together after 12 seconds. This 12 is the least common multiple (LCM) of 3 and 4.

The same thing happened in our problem. To find the time, when they will all toll together, we have to find the LCM of (2, 4, 8, 6, 10, 12).

LCM (2, 4, 8, 6, 10, 12) is  120

That is, 120 seconds or  2 minutes.

So, after every two minutes, all the bell will toll together.

For example, in 10 minutes, they toll together :

10/2 = 5 times

That is, after 2, 4, 6, 8, 10 minutes. It does not include the one at the start.

Similarly, in 30 minutes, they toll together :

(excluding one at the start).

Problem 5 :

The traffic lights at three different road crossings change after every 48 sec, 72 sec and 108 sec respectively. If they all change simultaneously at 8:20:00 hours, when will they again change simultaneously ?

For example, let the two signals change after every 3 secs and 4 secs respectively.

Then the first signal changes after 3, 6, 9, 12 seconds...

Like this, the second signal changes after 4, 8, 12 seconds...

So, if the two signals change simultaneously now, again they will change simultaneously after 12 seconds. This 12 is the least common multiple (LCM) of 3 and 4.

The same thing happened in our problem. To find the time, when they will all change simultaneously, we have to find the LCM of (48, 72, 108).

LCM (48, 72, 108) =  432 seconds  or  7 min 12 sec

So, after every 7 min 12 sec, all the signals will change simultaneously.

At 8:20:00 hours, if all the three signals change simultaneously, again they will change simultaneously after 7 min 12 sec. That is at 8:27:12 hours.

Hence, three signals will change simultaneously at 8:27:12 seconds.

Problem 6 :

Find the least number of soldiers in a regiment such that they stand in rows of 15, 20, 25 and form a perfect square.

To answer this question, we have to find the least number which is exactly divisible by the given numbers 15, 20 and 25. That is the least common multiple of (15, 20, 25).

LCM (15, 20, 25) = 300

So, we need 300 soldiers such that they stand in rows of 15, 20 , 25.

But, it has to form a perfect square (as per the question).

To form a perfect square, we have to multiply 300 by some number such that it has to be a perfect square.

To make 300 as perfect square, we have to multiply 300 by 3.

Then, it is 900 which is a perfect square.

Hence, the least number of soldiers required is 900.

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This is level 1: Finding the highest common factor (HCF) of two numbers.. You can earn a trophy if you get at least 9 correct and you do this activity online .

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Description of Levels

Sieve Use the Sieve of Eratosthenes to find prime numbers.

Factor Trees An interactive and very visual way to break down a number into its prime factors.

Level 1 - Finding the highest common factor (HCF) of two numbers.

Level 2 - Finding the lowest common multiple (LCM) of two numbers

Level 3 - Finding the highest common factor (HCF) of large numbers.

Level 4 - Finding the lowest common multiple (LCM) of large numbers

Level 5 - Finding the HCF and LCM of three numbers

Level 6 - Given the HCF and LCM find the numbers

Level 7 - Mixed application questions

HCF and LCM given An Advanced Lesson Starter.

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The highest common factor (HCF) of two numbers is the largest number that divides exactly into both of the numbers.

You can Find the HCF of numbers by listing the prime factors of both numbers then multiplying together the factors that appear in both lists.

For example find the HCF of 24 and 36 24 = 2x2x2x3 and 36 = 2x2x3x3 so the HCF of 24 and 36 is 2x2x3 = 12

The lowest common multiple (LCM), or least common multiple, is the smallest number that both numbers divide into exactly.

You can Find the LCM of numbers by listing the prime factors of both numbers and then multiply all the prime factors of the larger number by those prime factors of the smaller number that are not already included.

For example find the LCM of 24 and 36 24 = 2x2x2x3 and 36 = 2x2x3x3 so the LCM of 24 and 36 is 2x2x3x3 x 2 = 72

Venn Diagram

A Venn diagram may help you with the task of finding the HCF and LCM of 24 and 36.

Express each number as the primes which multiplied together would give you that number. Write them in Venn diagram sets:

Show the sets intersecting

Multiply the numbers in the intersection of the sets to find the HCF, 2x2x3 = 12.

Multiply all the numbers in the overlapping sets diagram to find the LCM, 2x2x2x3x3 = 72.

The Indian Method

Click here to see an animated demonstration of this cool way to find both the HCF and LCM of two numbers.

A Calculator Method

Advanced calculators have built in functions for finding the HCF and LCM of two numbers but there is a trick for finding the HCF using a modern scientific calculator.

If the two numbers are entered using the fraction template the calculator will express that fraction in its lowest terms. It does this by dividing numerator and denominator by their HCF.

For example to find the HCF of 24 and 36 enter 24/36 then press enter.

Considering the denominators, we now need to find what 24 was divided by to give 2. So dividing 24 by 2 gives 12 which is the HCF.

Connecting HCF and LCM

When you have found the HCF of the numbers a and b the LCM can be found using the following formula:

LCM = ab ÷ HCF

Greatest common divisor.

It is worth knowing that HCF is also known as GCD. If you are using a spreadsheet such as Excel there are functions named LCM and GCD for calculating the LCM and HCF.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly.

HCF and LCM exam questions

Understanding hcf and lcm.

• HCF stands for Highest Common Factor , which is the largest number that divides exactly into two or more given numbers.
• LCM denotes Lowest Common Multiple , referring to the smallest number that is a multiple of two or more given numbers.
• These concepts are fundamental in number theory and have applications in solving problems related to ratios, fractions, and divisibility.

Finding the HCF and LCM

• To find the HCF of two numbers, list all the factors of each number. The HCF is the highest number that appears in both lists.
• The LCM can be found by listing multiples of the numbers until you find a common value - this is your LCM. Alternatively, use the formula: LCM(a, b) = (a*b) / HCF(a,b).

HCF and LCM in Fractions

• The HCF is used when reducing fractions to their simplest form. The HCF of the numerator and denominator is used to divide both, giving the simplest form.
• The LCM is often used when adding or subtracting fractions with different denominators. The LCM of the denominators is used as the common denominator to make the calculation simpler.

Problem Solving with HCF and LCM

• Word problems often use HCF and LCM in context, such as organising events, scheduling, or planning. Recognising these types of problems and applying HCF and LCM correctly is key to finding solutions.

Common Mistakes to Avoid

• Mixing up the definitions of HCF and LCM. Always remember: HCF is the highest number that divides into the numbers, and the LCM is the smallest number that the numbers divide into .
• Overlooking factors or multiples when listing them out, resulting in an incorrect HCF or LCM.
• Forgetting to reduce fractions to their simplest form using the HCF.
• Not using the LCM as the common denominator when adding or subtracting fractions with different denominators.

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HCF and LCM word Problems, tricks & Solved examples

• The product of the two numbers is always equal to the product of their HCF and LCM.
• In case of HCF, if some remainders are given, then firstly those remainders are subtracted from the numbers given and then their HCF is calculated.
• In case of LCM, if a single remainder is given, then firstly the LCM is calculated and then that single reminder is added in that.
• In case of LCM, if for different numbers different remainders are given, then the difference between the number and its respective remainder will be equal. In that case, firstly the LCM is calculated, then that common difference between the number and its respective remainder is subtracted from that.
• Sometimes in case of HCF questions, the required remainder is given and when the remainder is not given, in those cases you will generally have three numbers given. For answering the question, you need to take the difference of the three pairs of numbers, now the HCF of these differences will become the answer e.g. if you have to find the greatest number, which when divides 83, 93 and 113 and leaves the same remainder. Here you will take the three differences i.e. 93 – 83 = 10 ; 113 – 93 = 20 ; 113 – 83 = 30, after that find the HCF of these differences, which comes out to be 10. Now you can check for yourself- when 10 divides these three numbers, the reminder obtained is 3 in each case and that is what the question was asking for.
• Whenever the question talks about the greatest or maximum, then in most of these cases it will be a question of HCF. Secondly, whenever the question is related to classification or distribution into groups, then in all the cases it is HCF only.
• Whenever the question talks about the smallest or minimum, then in most of the cases it will be a question of LCM. Secondly, whenever the word ‘together’ or ‘simultaneous’ is used in the question, then in all the cases it is LCM.
• Before solving the problems on HCF and LCM in the real exam, you must practice some HCF and LCM worksheets.
• HCF and LCM Concepts
• Problems, tricks & Solved examples
• HCF and LCM Problems (Basic)
• HCF and LCM Problems (Advanced)

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Typical problems on HCF and LCM | GCD and LCM Problems & Solutions

Practice problems on hcf and lcm for all competitive exams.

Common Divisor : A number which exactly divides all the given numbers is a common divisor of those numbers.

For example ” 3″ is c ommon divisor of 9, 12, 15, 18.

Greatest Common Divisor(GCD) or Highest Common factor ( HCF )  : The greatest number that exactly divides all the given number is the GCD or HCF of those numbers

Common Multiple : A number which is exactly divisible by all the given numbers is a common multiple of those numbers.

For example ” 15 ” is a common multiple of 3 and 5.

Least Common Multiple (LCM) : The lowest number which is exactly divisible by all the given numbers is the LCM of those numbers.

The concepts and methods to find L owest Common Multiple (LCM) and Highest Common Factor (HCF ) explained in the below link

Concepts of Highest Common Factor (Greatest common divisor –  GCD) and Lowest Common Multiple | Factorization and Division Methods for finding HCF and LCM.

In this page explained different types  of typical problems with solutions on HCF ( GCD ) & LCM

Some Important tips in HCF and LCM.

a) Find the least number which when divided by n1 , n2 & n3 leaves the remainders x ,y , and z respectively. Here answer is [ LCM of (n1, n2, n3) – k ] where k =  (n1 -x) = (n2 -y) = (n3 -z).

 b) Find the least number which when divided by n1 , n2 & n3 leaves the same remainder ” k” . Here answer is [ LCM of (n1, n2, n3) + k ]

c) Find the greatest number which exactly divide  n1 , n2 & n3 leaves the same remainder ” k” . Here answer is [ HCF of (n1-k, n2-k, n3-k).

d) Find the greatest number which exactly divide  n1 , n2 & n3 leaves the same remainder  x, y and z respectively . Here answer is [ HCF of (n1-x, n2-y, n3-z).

e) GCD (m, n ) x LCM(m, n) = m x n ( i.e The product of HCF and the LCM equals the product of the numbers.

f) HCF of fractions and LCM of fractions.

Example – 1 : What is the side of largest possible square slab which can be paved on the floor of a room 2m 73cm long and 3m 25cm broad?

Sol : Here possible largest square slab means it equal to greatest common divisor of both sides i.e 2m 73cm and 3m 25cm.

i.e 2m 73cm = 273cm & 3m 25cm = 325cm. Now find the GCD for 273 and 325.

273 = 13 1 x 7 1 x 3 1

325 = 13 1 x 5 2

GCD of 273 and 325 is 13.

So 13cm is the side of maximum possible square slab.

Example – 2 : There are 576 boys and 448 girls in a school that are to be divided into equal sections of either boys or girls alone. Fine the minimum total number of sections thus formed.

Sol : Here find height common factor for numbers 576 and 448

i.e 576 = 18 x 32 = 3 2 x 2 6

448 = 2 6 x 7 1

Highest Common Factor ( HCF ) = 2 6 = 64.

So minimum number of sections formed with boys and girls alone = (576 / 64) + (448/64)

= 7 + 9 = 16 sections.

Example – 3 : Find the greatest number of 5 digits, that will gives us a remainder of 5, when divided by 8 and 9 respectively?

Sol: First calculate LCM of 8 and 9 = 8 x 9 = 72 ( due to not having the common divisors).

Now find the largest five digit multiple number of 72 .

i.e 99936 ( 99999 /72 = 1388._ _  so 1388 x 72 = 99936 )

Our answer is 99936 + 5 = 99941.

Example – 4 : Find the greatest number of four digits which when divided by 10, 11, 15 and 22 leaves 3 , 4 , 8 and 15 as remainders respectively?

Sol: First calculate LCM of 10, 11, 15 and 22 =  330

Now find the largest five digit multiple number of 330 .

i.e 9900( 9999/330 = 30._ _  so 30 x 330 = 9900 )

Our answer is 9900 -7 = 9893.

Example – 5 : Find the smallest number which when divided by 13 and 16 leaves respective remainder of 2 and 5?

Solution : Here first identify  (13 – 2) = (16 – 5) = 11.

Now find the LCM of 13 and 16 = 13 x  16 = 208.

Our final answer is 208 – 11 = 197.

Example – 6 : What is the least number when divided by 5, 6 , 8 , 10 leaves 2 as the remainder in each case?

Sol: LCM of 5, 6, 8, 10 = LCM of 6, 8 , 10 = 120.

Our final answer is 120 + 2 = 122.

Example – 7 : Find the least number that when divides by 16, 18 and 20 leaves a remainder 4 in each case , but is completely divisible by 7

Sol: First find the Least Common Multiple ( LCM ) of 16 , 18 and 20.

i.e 16 = 2 4

18 = 3 2 x 2 2

20 = 5 1 x 2 2

LCM of 16 , 18 and 20  = 2 4 x  32 x 5 1   = 720.

Now here leaves the remainder 4 in each case and should be divisible by 7

Take 720 + 4 = 724 ( 7 is not divisor of 724 )

Now go for next number 720 + 720 + 4 = 1444 ( 7 is not divisor of 1444 )

Again go for next number 720 + 720 + 720 + 4 = 2164 ( 7 is not divisor of  2164 )

Again go for next number 720 + 720 + 720 + 720+ 4 = 2884 ( 7 is divisor of  2884 )

Hence our final answer is 2884.

Example – 8 : Find the least Multiple of 7 which leaves a remainder of 4 when divided by 6, 9 15 and 18 ?

Sol : It can be find  like above example

i.e LCM of 6 , 9 , 15  and 18 = 180.

Now find the series of LCM wit h remainder having 4

i. e 184, 364,  …..  ( Here 7 is divisor of 364 )

So our answer is 364.

Example – 9 : Three bells ring at intervals of 5 seconds, 6seconds and 7 seconds respectively. If they toll together for the first time at 9AM in the morning, after what interval of time will they together ring again for thee first time.

Sol: Here we find Least Common Multiple (LCM) of 5, 6 & 7

i.e 5 x 6 x 7 = 210 ( here not having any common divisors in between two number out of all these numbers  expect “1”. So multiplying all these numbers to get lcm)

Here our answer is 210 seconds.

Example – 10 : Find the LCM and GCD of 4/5, 5/6, 7/15 ?

Sol: Find the LCM of numerators (4 , 5 , 7 ) = 4 x 5 x 7 = 140 ( here not having any common divisors in between two number out of all these numbers  expect “1”. So multiplying all these numbers to get lcm)

GCD of numerators (4 , 5 , 7 ) = 1  ( Not having common divisors except 1 )

LCM of denominators  (5, 6, 15 ) = 5 1 , 3 1 x 2 1 , 3 1 x5 1 =  5 x 3 x 2 = 30

GCD of denominators  (5, 6, 15 ) =  1  ( Not having common divisors except 1 )

Using formulas

and GCD of 4/5, 5/6, 7/15 = 1 / 30.

Example – 11 : The GCD and LCM of two numbers are 66 and 384. If one of the numbers divided by 2 gives the results as 192, what is the second number?

Sol : It is very simple find the first numbers i.e 192 x 2 = 384.

Now we know LCM x GCD = 384 x n2

So n2 ( second number) = 66.

Some related Topics in Quantitative aptitude

The Concepts of number system the mathematics

Divisibility Rules of numbers from 1 to 20 | Basic math education

Simple interest and Compound interest formulas with examples

Percentage formulas | percentage calculations with examples

Circle formulas in math | Area, Circumference, Sector, Chord, Arc of Circle

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Types of Triangles With examples | Properties of Triangle

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6 thoughts on “ Typical problems on HCF and LCM | GCD and LCM Problems & Solutions ”

hamisi machambula

it better at all

It’s a dumb ass

shahin akter

if hcf of two numbers is 12.sum of two numbers 72..find the two numbers

Thank you Nani

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HCF and LCM - Question Page

Find the Higest Common Factor:

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• Math Article
• Hcf And Lcm

HCF and LCM

The full form of LCM in Maths is the Least Common Multiple, whereas the full form of HCF is the Highest Common Factor. The H.C.F. defines the greatest factor present in between given two or more numbers, whereas L.C.M. defines the least number which is exactly divisible by two or more numbers. H.C.F. is also called the greatest common factor (GCF) and LCM is also called the Least Common Divisor.

To find H.C.F. and L.C.M., we have two important methods which are the Prime factorization method and the division method. We have learned both methods in our earlier classes. The shortcut method to find both H.C.F. and L.C.M. is a division method. Let us learn the relationship between HCF and LCM with the help of the formula here. Also, we will solve some problems based on these two concepts to understand in a better way. The article here is very helpful for primary and secondary classes students such as Class 4, Class 5, Class 6, Class 7, and Class 8.

Table of Contents:

HCF Definition

Lcm definition, lcm of two numbers, hcf and lcm formula, hcf by prime factorization method, hcf by division method, lcm by prime factorization method, lcm by division method, hcf and lcm examples, practice problems, hcf and lcm definition.

We know that the factors of a number are exact divisors of that particular number. Let’s proceed to the highest common factor (H.C.F.) and the least common multiple (L.C.M.).

The full form of HCF in Maths is Highest Common Factor.

As the rules of mathematics dictate, the greatest common divisor or the gcd of two or more positive integers happens to be the largest positive integer that divides the numbers without leaving a remainder. For example, take 8 and 12. The H.C.F. of 8 and 12 will be 4 because the highest number that can divide both 8 and 12 is 4.

The full form of LCM in Maths is Least Common Multiple.

In arithmetic, the least common multiple or LCM of two numbers say a and b, is denoted as LCM (a,b). And the LCM is the smallest or least positive integer that is divisible by both a and b. For example, let us take two positive integers 4 and 6.

Multiples of 4 are: 4,8,12,16,20,24…

Multiples of 6 are: 6,12,18,24….

The common multiples for 4 and 6 are 12,24,36,48…and so on. The least common multiple in that lot would be 12. Let us now try to find out the LCM of 24 and 15.

More is here: Learn Mathematics

Suppose there are two numbers, 8 and 12, whose LCM we need to find. Let us write the multiples of these two numbers.

8 = 16, 24, 32, 40, 48, 56, …

12 = 24, 36, 48, 60, 72, 84,…

You can see, the least common multiple or the smallest common multiple of two numbers, 8 and 12 is 24.

The formula which involves both HCF and LCM is:

Say, A and B are the two numbers, then as per the formula; A x B = H.C.F.(A, B) x L.C.M.(A, B) We can also write the above formula in terms of HCF and LCM, such as: H.C.F. of Two numbers = Product of Two numbers/L.C.M of two numbers & L.C.M of two numbers = Product of Two numbers/H.C.F. of Two numbers

NOTE-  The above relation between H.C.F and L.C.M is not valid for the product of numbers greater than 2. It is only valid for the product of two numbers.

How to find HCF and LCM?

Here are the methods we can use to find the HCF and LCM of given numbers.

• Prime factorization method
• Division method

Let us learn both methods, one by one.

Take an example of finding the highest common factor of 144, 104 and 160. Now let us write the prime factors of 144, 104 and 160. 144 = 2 × 2 × 2 × 2 × 3 × 3 104 = 2 × 2 × 2 × 13 160 = 2 × 2 × 2 × 2 × 2 × 5 The common factors of 144, 104 and 160 are 2 × 2 × 2 = 8 Therefore, HCF (144, 104, 160) = 8

Steps to find the HCF of any given numbers;

Hence, we can see here that 16 is the highest number which divides 160 and 144. Therefore, HCF (144, 160) = 16

To calculate the LCM of two numbers 60 and 45. Out of other ways, one way to find the LCM of given numbers is as below:

• List the prime factors of each number first. 60 = 2 × 2 x 3 × 5 45 = 3 × 3 × 5
• Then multiply each factor the most number of times it occurs in any number.

If the same multiple occurs more than once in both the given numbers, then multiply the factor by the most number of times it occurs. The occurrence of Numbers in the above example: 2 : two times 3 : two times 5 : one time LCM = 2 × 2 x 3 × 3 × 5 = 180

Therefore, LCM of 60 and 45 = 2 × 2 x 3 × 3 × 5 = 180 At BYJU’S you can also learn,  Prime Factorization Of Hcf And Lcm .

Find the Highest Common Factor of 25, 35 and 45. Solution:

Given, three numbers as 25, 35 and 45.

We know, 25 = 5 × 5

From the above expression, we can say 5 is the only common factor for all three numbers.

Therefore, 5 is the HCF of 25, 35 and 45.

Find the Least Common Multiple of 36 and 44.

What is the L.C.M. of 25, 30, 35 and 40?

L.C.M. of 25, 30, 35 and 40

Let us find LCM by prime factorisation.

Prime factorisation of 25 = 5 x 5 = 5 2

Prime factorisation of 30 = 2 x 3 x 5 

Prime factorisation of 35 = 5 x 7

Prime factorisation of 40 = 2 x 2 x 2 x 5 = 2 3 x 5

LCM (25, 30, 35, 40) = 2 x 2 x 2 x 3 x 5 x 5 x 7 = 4200

The HCF of the two numbers is 29 & their sum is 174. What are the numbers?

Let the two numbers be 29x and 29y. 

Given, 29x + 29y = 174 

29(x + y) = 174 

x + y = 174/29 = 6

Since x and y are co-primes, therefore, possible combinations would be (1,5), (2,4), (3,3).

The only combination that follows the co-prime part is (1,5)

For (1,5): 29 x = 29 x 1 and 29 y = 29 (5) = 145

Therefore, the required numbers are 29 and 145.

Find the product of two numbers whose H.C.F is 25 and L.C.M is 5.

Product of two numbers = H.C.F x L.C.M

In this problem, the product is not possible as H.C.F given in the problem is greater than the L.C.M which is not possible.

• Find the HCF of 43, 91 and 183.
• Which is the greatest number of four digits which is divisible by 15, 25, 40 and 75?
• Three numbers are in the ratio of 3: 4: 5 and their L.C.M. is 2400. What is the value of HCF?
• What is the LCM of 24, 36 and 40?

Frequently Asked Questions on HCF and LCM

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Problem on HCF and LCM

Question 1: Find the HCF by long division method of two no’s the sequence of quotient from top to bottom is 9, 8, 5 and the last divisor is 16. Find the two no’s. Solution: Start with the divisor and last quotient. Divisor x quotient + remainder = Dividend 16 x 5 + 0 = 80 80 x 8 + 16 = 656 656 x 9 + 80 = 5984 Hence, two numbers are 656 and 5984.

Question 2: The LCM and HCF of two numbers is 210 and 5. Find the possible number of pairs. Solution: HCF = 5 so it should be multiple of both numbers. So both numbers 5x : 5y LCM = 5 * x * y = 210 x * y = 42 {1 x 42}, { 2 x 21}, {3 x 14}, { 6 x 7 } . Four pairs are possible .

Question 3: The sum of two numbers is 132 and their LCM is 216. Find both the numbers. Solution:

Note: HCF of Sum & LCM is also same as actual HCF of two numbers. Factorize both 132 and 216 and find the HCF. 132= 2 2 x 3 x 11 216= 2 3 x 3 3 HCF= 2 2 x 3 =12

Now, 12x + 12y = 132 x + y = 11 And 12 * x * y = 216 x * y = 18 Solve for x and y, we get y = 9 and x = 2. Hence both numbers are 12*2 = 24 and 12*9 = 108

Question 4: The LCM of two numbers is 15 times of HCF. The sum of HCF and LCM is 480. If both number are smaller than LCM. Find both the numbers. Solution: LCM = 15 * HCF We know that LCM + HCF = 480 16 * HCF = 480 HCF = 30 Then LCM = 450 LCM = 15 HCF 30 * x * y = 15 * 30 x * y = 15 Factors are {1 x 15} and { 3 x 5} Both numbers less than LCM so take {3 x 5} Hence numbers are 3 * 30 = 90 and 5 * 30 = 150

Question 5: Find the least perfect square number which when divided by 4, 6, 7, 9 gives remainder zero. Solution: Find the LCM for 4, 6, 7, 9 LCM= 2 2 * 3 2 * 7 = 252 To become perfect square all factors should be in power of 2. So, multiply it by 7 LCM = 2 2 * 3 2 * 7 2 = 1764 And it is perfect square of 42 .

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HCF and LCM problem solving

Subject: Mathematics

Age range: 14-16

Resource type: Worksheet/Activity

Last updated

30 December 2019

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Questions where you are given the HCF and LCM and have to work out what the possibe numbers could be.

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