Algebra and General Mathematics: Factors

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 (x² – y²)

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 2⁴

F₍₆₀₀₎  = 3 x 5² x 2³

F₍₇₂₀₎  = 3² x 5 x 2⁴

Therefore, the HCF is:

HCF = 3 x 5 x 2³ = 120

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