# Find the LCM and the GCF of Integers - Examples and Questions with Answers (Grade 5)

How to find the LCM and GCF of integers? Examples and questions with answers are presented. This is a tutorial on how to find the LCM and GCF. In order to help you check your answers, a calculator for the LCM and a calculator for the GFC of two numbers are provided.
Lowest Common Multiple (LCM) Calculator. Calculate the lowest common multiple of two positive integers.
Greatest Common Factor (GCF) Calculator. Calculate the greatest common factor of two positive integers.

## A - Find the Lowest Common Multiple (LCM) of two Numbers

Method 1:
Example 1:
Find the LCM of 6 and 4 listing the multiples.
Find the first few multiples of the two numbers 6 and 4 as follows:
6: 6 , 12 , 18 , 24 , 30 ,...
4: 4 , 8 , 12 , 32 , 40 ,...
The lowest multiple that is common to 6 and 4 is 12. So the LCM of 6 and 4 is 12.
The above method works well for small numbers only.

Method 2:
Find the LCM using prime factorization.
Prime factorization uses the prime numbers 2, 3, 5, 7, 11, ... to factor an integer.
Examples of Prime Factorization:
4 = 2 × 2
14 = 2 × 7
16 = 2 × 2 × 2 × 2
20 = 2 × 2 × 5

Example 2:
Find the LCM of 6 and 4 using prime factorization.
Find the prime factorization of the two numbers.
6 = 2 × 3
4 = 2 × 2
A common multiple may be found by multiplying the two numbers 6 and 4. However it may not be the lowest. Let us multiply the two numbers in factored form :
6 × 4 = (2 × 3) × ( 2 × 2)
One factor 2 is counted twice and therefore has to be taken out of one term of the product if we want our common multiple to be the lowest.
The LCM of 4 and 6 = 3 × (2 × 2) = 12

Example 3:
Find the LCM of 20 and 24 using prime factorization.
Find the prime factorization of the two numbers.
20 = 2 × 2 × 5
24 = 2 × 2 × 2 × 3
The product of 20 and 24 is given by (2 × 2 × 5) × (2 × 2 × 2 × 3)
2 x 2 is counted twice and therefore has to be taken out of one term and so the LCM is given by
LCM = (5) × (2 × 2 × 2 × 3) = 120

Example 4: Find the LCM of 1240 and 5300 using prime factorization.
Find the prime factorization of the two numbers.
1240 = 2 × 2 × 2 × 5 × 31
5300 = 2 × 2 × 5 × 5 × 53
The product of 1240 and 5300 is given by (2 × 2 × 2 × 5 × 31) × (2 × 2 × 5 × 5 × 53)
2 × 2 × 5 is counted twice and therefore has to be taken out of one term and so
LCM of 1240 and 5300 = ( 2 × 31) × (2 × 2 × 5 × 5 × 53) = 328,600

## A - Find the Greatest Common Factor (GCF) of two Numbers

Example 5:
Find the GCF of 6 and 4 using prime factorization.
6 = 2 × 3
4 = 2 × 2
2 is a common factor to both 6 and 4 and it is the highest.
So the GCF of 6 and 4 is equal to 2.

Example 6:
Find the GCF of 20 and 24 using prime factorization.
20 = 2 × 2 × 5
24 = 2 × 2 × 2 × 3
Underline all common factors in the factorization of 20 and 24.
20 = 2 × 2 × 5
24 = 2 × 2 × 2 × 3
The greatest common factor of 20 and 24 is the product of all common factors 2 × 2 = 4.

Example 7:
Find the GCF of 1240 and 5300 using prime factorization.
1240 = 2 × 2 × 2 × 5 × 31
5300 = 2 × 2 × 5 × 5 × 53
Underline common factors
1240 = 2 × 2 × 2 × 5 × 31
5300 = 2 × 2 × 5 × 5 × 53
The GCF of 1240 and 5300 is given by 2 × 2 × 5 = 20

## C - Relationship Between The LCM and the GCF of two numbers

The LCM's and GCF's found above are organized in the table. It is shown that the product of any two whole numbers is equal to the product of their LCM and GCF.

(m , n) LCM(m,n) GCF(m,n) LCM(m,n) × GCF(m,n) m × n
(4 , 6) 12 2 12 × 2 = 24 4 × 6 = 24
(20 , 24) 120 4 120 × 4 = 480 20 × 24 = 480

Hence the property: For any two integers m and n, we have the relationship between the product m × n and the product of their LCM and GCF as follows:
m × n = LCM(m,n) × GCF(m,n)

You may use these calculators
Lowest Common Multiple (LCM) Calculator and Greatest Common Factor (GCF) Calculator to investigate the above property with more examples.

## Questions

Find the LCM and GCF of each pair of numbers and check that the product of the two numbers is equal to the product of the LCM and the GCF.

1. 21 , 14
2. 45 , 55
3. 120 , 248

## Answers to the Above Questions

1. GCF = 7 , LCM = 42 , GCF × LCM = 294 , m × n = 21 × 14 = 294
2. GCF = 5 , LCM = 495 , GCF × LCM = 2475 , m × n = 45 × 55 = 2475
3. GCF = 8 , LCM = 3720 , GCF × LCM = 29760 , m × n = 120 × 248 = 29760

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