Factors And Multiples

If a is divisible by b, then b is a factor of a, and a is a multiple of b.

For example, 30 = 3 × 10, so 3 and 10 are factors of 30 and 30 is a multiple of 3 and 10

Take note that 1 is a factor of every number.

Understanding factors and multiples is essential for solving many math problems.

Prime Factors

A factor which is a prime number is called a prime factor.

For example, the prime factorization of 180 is 2 × 2 × 3 × 3 × 5

You can use repeated division by prime numbers to obtain the prime factors of a given number.

Greatest Common Factor (GCF)

As the name implies, we need to list the factors and find the greatest one that is common to all the numbers.

For example, to get the GCF of 24, 60 and 66:

The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24
The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60
The factors of 66 are 1, 2, 3, 6, 11, 22,33 and 66

Look for the greatest factor that is common to all three numbers - thus 6 is the GCF of 24, 60 and 66.

Least Common Multiple (LCM)

As the name implies, we need to list the multiples and to find the least one that is common to all the numbers.

For example, to get the LCM of 3, 6 and 9:

The multiples of 3 are 3, 6, 9, 12, 15, 18, 21 ...
The multiples of 6 are 6, 12, 18, 24, ...
The multiples of 9 are 9, 18, 27, ...

Look for the least multiple that is common to all three numbers - thus 18 is the LCM of 3, 6 and 9.

Shortcut To Finding LCM
Here is a useful shortcut to finding the LCM of a set of numbers. For example, to find the LCM of 3, 6 and 9, we divide them by any factor of the numbers in the following manner:

The following video highlights the difference between greatest common factor and least common multiple.

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