Lowest common multiple (LCM)

The Least Common Multiple (LCM) is also referred to as the Lowest Common Multiple (LCM) and Least Common Divisor (LCD). For two integers a and b, denoted LCM(a,b), the LCM is the smallest positive integer that is evenly divisible by both a and b. For example, LCM (3,5) = 10 and LCM(7,8) = 56.

 

How to find LCM

1        Listing Multiples

• List the multiples of each number until at least one of the multiples appears on all lists
• Find the smallest number that is on all of the lists
• This number is the LCM

Example: LCM (6,7,21)

• Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
• Multiples of 7: 7, 14, 21, 28, 35, 42, 56, 63
• Multiples of 21: 21, 42, 63

Find the smallest number that is on all of the lists. We have it in bold red above.

So LCM(6, 7, 21) is 42

2        By prime Factorization

• Create a set of prime factors of each given number.
• Create a union set for the factors.
• The products of union set of prime factors is the LCM

Example 1

Find LCM of 9 and 12

A set A of prime factors of 12 = {2, 2, 3)

A set B of prime factors of 9 = {3, 3)

A U B  = {2, 2, 3,3}

LCM of 9 and 12 = 2 x 2 3 x 3 = 36

Or use a table to find the prime factors

2	9	12
2	9	6
3	3	3
3	1	1

LCM = 2 x 2 x 3 x 3 =36

Example 2

There are three classes in a school with 4, 15 and 18 pupils respectively. Find the number of mangoes that each class can share out without any remainders.

Solution

The questions requires to find the LCM of 4, 15 and 18

2	4	15	18
2	2	15	9
2	1	15	9
3	1	5	3
3	1	5	1
5	1	1	1

LCM = 2 x 2 x 3 x 3 x 5 = 180

 

Greatest common Factor (GCF)

This is found by finding the products of product of the intersection set of prime factors of the numbers.

It can be obtained by

1.      Listing Factors and identify the biggest common factor in the lists.

Example 3

Find the greatest common factor, GCF, of 36 and 63.

Factors of 36 = {1, 3, 6, 9, 18, 36}

Factors of 63 = {1, 3, 7, 9, 21, 63}

The greatest common factor = 9 (in bold)

2. By Finding the product of the elements of intersection set of prime factors of numberrs.

Example 4

What is the greatest common factor (GCF) of 36 and 63?

Set A = prime factors of 36 or {2, 2, 3, 3}

Set B = prime factors of 63 or {3, 3, 7}

A ∩B = {3, 3)

Therefore, GCF = 3 x 3 = 9

For additional exercise and answers download PDF below

Lowest Common Multiple (LCM) and Greatest Common Factor (GCF) – upper primary

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