Factors and multiples of numbers

Consider two numbers 5 and 6. When we multiply these numbers (5 × 6), the answer is 30. The numbers 5 and 6 are called factors (or divisors) of 30, and the number 30 is called a multiple of 5 and 6. Therefore, factors are the divisors of a number.

Example

Find all factors of 12.

Note that when listing the factors, we don't repeat any of them.

Factors of 12 are: 1, 2, 3, 4, 6, 12

Multiples of a number are the products of multiplying that number by natural numbers. For example, multiples of 4 are 4, 8, 12, 16, 20, 24, …. This means multiply 4 by 1, 2, 3, 4, 5, 6, ….

Example

List all multiples of 6 between 30 and 45.

The multiples of 6 are: 6 × 1 = 6; 6 × 2 = 12; 6 × 3 = 18; 6 × 4 = 24; 6 × 5 = 30; 6 × 6 = 36; 6 × 7 = 42; 6 × 8 = 48 and so on.

Therefore, multiples of 6 between 30 and 45 are 36 and 42.

The Greatest Common Factor is sometimes called the Highest Common Factor. Its short form is GCF or HCF respectively. The Greatest Common Factor is the largest common divisor of two or more numbers given.

For example, if you are told to find the GCF of 15 and 25, first list all factors or divisors of 15 and that of 25. Thus, factors of 15 are 1, 3, 5, 15; factors of 25 are 1, 5, 25. The common factors are 1 and 5. Therefore the GCF is 5.

Example

Find the HCF of 72 and 120.

We have to list factors of our numbers:

Factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Factors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

The common factors are: 1, 2, 3, 4, 6, 8, 12, 24

Therefore, the HCF of 72 and 120 is 24.

Method 2: Prime Factorization

Example

Find the HCF of 36 and 48.

We have to find prime factors of 36 and 48 first. Thus:

$36 = 2 \times 2 \times 3 \times 3$

$48 = 2 \times 2 \times 2 \times 2 \times 3$

After writing the numbers as a product of their prime factors, take only the common prime factors (prime factors that appear in all numbers). In our example, the common factors are $2 \times 2 \times 3$. Therefore, the HCF of 36 and 48 is 12.

Lowest Common Multiple is also called Least Common Multiple, and its short form is LCM.

For example: multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, …; multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, ….

When you look carefully at these multiples of 5 and 3, you notice that 15 appears in both. This multiple that appears in both is called a common multiple. If there is more than one common multiple that appears, the smallest common multiple is called the Lowest Common Multiple.

Method 1: Listing Multiples

Example

Find the common multiples and then show the Lowest Common Multiple of the numbers 4, 6, and 8.

Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, …

Multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …

Multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, 72, …

The common multiples of 4, 6, and 8 are 24, 48, …

Therefore, the Lowest Common Multiple of 4, 6, and 8 is 24.

Method 2: Prime Factorization

Example

Find the LCM of 24 and 36 by prime factorization.

Let us find prime factors of each number by dividing the numbers by their prime factors.

$24 = 2 \times 2 \times 2 \times 3$

$36 = 2 \times 2 \times 3 \times 3$

Now take the prime factors that appear in both numbers (we take without repeating). We are left with 2 (which is a factor of 24) and 3 (which is a factor of 36). We also have to multiply these prime factors left: $2 \times 2 \times 2 \times 3 \times 3$. This gives 72.

Therefore, the LCM of 24 and 36 is $2 \times 2 \times 2 \times 3 \times 3 = 72$.