#### FACTORS

A factor is a number or expression which divides exactly another number or expression.

Example:

2 and 4 are factors of 8

(x + y) and (x – y) are factors of (x^{2} – y^{2})

Note: the **Proper Factors** of a number are its entire factor except the number itself.

Example:

Factors of 24, F_{(24) }= { 1, 2, 3, 4, 6, 8, 12, 24 }

Proper Factors, F_{p} = { 1, 2, 3, 4, 6, 8, 12 }

#### COMMON FACTORS

Common Factors are factors shared or common to the two or more numbers.

Example:

What are the common factors of 12 and 24?

Solution:

Factors of 12:

F_{(12) }= { 1, 2, 3, 4, 6, 12 }

Factors of 24:

F_{(24) }= { 1, 2, 3, 4, 6, 8, 12, 24 }

Therefore, the common factors of 12 and 24 are:

**F _{(12)} ⋂ F_{(24)} = { 1, 2, 3, 4, 6 }**

#### LOWEST COMMON FACTORS (LCF)

The Lowes Common Factor (LCF) of two or more numbers is the common factor of all those numbers which has the lowest value.

Example: ECE Board April 1998

What is the lowest common factor of 10 and 32?

Solution:

Prime factors of 10 = 5 x 2

Prime factors of 32 = 2 x 2 x 2 x 2 x 2

**F _{2⋂32} = 2**

#### HIGHEST COMMON FACTORS (HCF)

The Highest Common Factor (HCF) of two or more whole numbers is the largest whole number which is a factor of both.

Example:

Find the HCF of 240, 600, and 720.

Solution:

-get all the prime factors of the numbers in canonical form

F_{(240)} = 3 x 5 x 2^{4}

F_{(600)} = 3 x 5^{2} x 2^{3}

F_{(720)} = 3^{2} x 5 x 2^{4}

Therefore, the HCF is:

**HCF = 3 x 5 x 2 ^{3} = 120**

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