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Definition:

A factor of a number is always an exact divisor of that number. For eg: 4 is a factor of 24. This means that the number 24 is exactly divisible by 4 (Remainder of this division is 0). Also note that a factor of a number is always less than or equal to that number. For example: The factors of 12 are 1, 2, 3, 4, 6, 12. Each of these factors is less than or equal to the number 12.

Factors exist in pairs. For example: the factors of 18 are 1, 2, 3, 6, 9, 18. These factors can be paired as follows: 1 & 18, 2 & 9, 3 & 6. The product of each pair is 18. These pairs of factors are referred to as factor pairs.

Practice questions:

Q1: List the factors of 60.

A1: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

• Every number is a factor of itself.

For example: 24 x 1 = 24, hence 24 is a factor of itself

• 1 is a factor of every number

• Prime numbers have only two factors - the number itself and the number 1.

For example: the factors of 17 are 17 and 1.

• The number of factors of a given number are always finite.

For example: the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

Common Factors

Consider the factors of 24 and factors of 32

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Factors of 32: 1, 2, 4, 8, 16, 32

If you notice there are some numbers that are factors of both 24 and 32  namely 1, 2, 4, 8. Such numbers that are factors of two or more numbers are referred to as common factors of those numbers.

Common factors can also be represented by Venn diagrams.

Highest Common Factor (HCF)

In the above example, the number 8 is the highest common factor to appear in both lists and is called the Highest Common Factor of 24 and 32. Highest Common Factor or HCF is sometimes also referred to as the Greatest Common Divisor.

The Highest Common Factor of two or more numbers is defined as the greatest number that divides each of those numbers exactly.

Practice questions:

Q1: List the common factors of 20 and 18.

A1: Factors of 20: 1, 2, 4, 5, 10, 20

Factors of 18: 1, 2, 3, 6, 9, 18

The common factors of 20 and 18 are 1 and 2.

Q2: Find the Highest Common Factor of 24 and 64.

A2: The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.

Factors of 64: 1, 2, 4, 8, 16, 32, 64.

The common factors of 24 and 64 are 1, 2, 4, 8. The HCF of 24 and 64 is 8.

Q3: Find the Highest Common Factor of 13 and 21.

A3: The factors of 21 are 1, 3, 7, 21

Factors of 13: 1, 13 (Note that 13 is a prime number)

The Highest Common Factor of 13 and 21 is 1.

Important facts about Highest Common Factors:

• The HCF of a given set of numbers can never be greater than any of the numbers in the set.

• Co-prime numbers are numbers who do not have any common factor between them other than 1. In other words, the HCF of co-prime numbers is always 1.

• The product of HCF and LCM of two numbers is equal to the product of those numbers.

For example: Consider two number 12 and 9.

The LCM of 12 and 9 is 36.

The HCF of 12 and 9 is 3.

The Product of 12 and 9 is 108.

The product of LCM and HCF is 36 x 3 = 108.

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