# HIGHEST COMMON FACTOR

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INTRODUCTION

In mathematics, the Highest Common Factors (HCF), also known as the Greatest common factor (GCF), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. For example, the HCF of 8 and 12 is 4.

This notion can be extended to polynomials.

In algebra, the Highest common factor (frequently abbreviated as HCF) of two polynomials is a polynomial, of the highest possible degree, that evenly divides each of the two original polynomials. This concept is analogous to the highest common divisor of two integers.

In algebra, the Highest common divisor (frequently abbreviated as HCF) of two polynomials is a polynomial, of the highest possible degree, that evenly divides each of the two original polynomials. This concept is analogous to the highest common divisor of two integers.

CHAPTER # 1

FACTORIZATION

Factorization mean of splitting a number or polynomial in its components,  factors of a number or polynomial are those that divide the number with zero remainder, for example

Factorization of digits

Factors of  9 , 12,         6 and 18

9:      factors  are 1,3,9

12:   factors  are  1,2,3,4,6,12

6:      factors  are           1,2,3,6

18:     factors  are     1,2,3,6,9,18

Factors of 15 are 1, 3, 5, and 15

Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30

Factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105

CHAPTER # 2

FACTORIZATION OF POLINOMIALS

There are different ways of finding the factors of a polynomial depend on the situation some of them are discuss below

by taking common K method e.g.

Kx+Ky+K=K(x+y+1) here K is factor

By head to tail method eg

4x2+7x+3=(x=1)(4x+3)

by squaring formula i.e (a+b)2,(a-b)2   eg:

4x2+4x+1=(2x+1)(2x+1)

4x2-4x+1=(2x-1)(2x-1)

by sum and difference formula i.e a2-b2,a3+b3,a3-b3

a2-b2=(a+b)(a-b) eg:

9x2-16y2=(3x-4y)(3x+4y)

a3+b3=(a+b)(a2-ab+b2) eg:

27x3+8y3=(3x+2y)(9x-6xy+4y)

a3-b3=(a-b)(a2+ab+b2) eg:

27x3-8y3=(3x-2y)(9x+6xy+4y)

by cube formula (a+b)3, (a-b)3

(a+b)3=(a+b)(a+b)(a+b) eg:

(ax+by)3=(ax+by)(ax+by)(ax+by)

(a-b)3 =(a-b)(a-b)(a-b) eg:

(3a-2b)3=(3a-2b)(3a-2b)(3a-2b)

By remainder theorem  eg:

x2 + 7x + 6 = (x + 1)(x + 6)

x2 − 5x − 6 = (x + 1)(x − 6)

By formula of the forms (a+b+c)2 and a3+b3+c3-3abc

(a+b+c)2=(a+b+c)(a+b+c)

a3+b3+c3-3abc=(a+b+c)(a2+b2+c2-ab-bc-ca)

CHAPTER # 3

HIGHEST COMMON FACTOR OF DIGITS

The "Highest Common Factor" is the largest of the common factors (of two or more numbers)

Finding the Highest Common Factor

Here are three ways:

•             You can:

find all factors of both numbers

then select the ones that are common to both, and

then choose the Highest.

Example:

Two Numbers           Factors             Common Factors          Highest

Common Factor Example Simplified

Fraction: 9 and 12

9:                         1,3,9                                  1,3                                 3

12:                         1,2,3,4,6,12                              1,3

9/12 = 3/4 which dived both numbers

And another example:

Two Numbers         Factors               Common Factors           Highest

Common Factor Example Simplified

Fraction   6 and 18

6:                         1,2,3,6

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

6/18 = 1/3 which dived both numbers

2. You can find the prime factors and combine the common ones together:

Two Numbers                                                   Thinking ...          Highest

Common Factor Example Simplified

Fraction

24 and 108

2 × 2 × 2 × 3 = 24, and                                                                                 12

2 × 2 × 3 × 3 × 3 = 108                              2 × 2 × 3 = 12

24/108 = 2/9   which dived both numbers

3. And sometimes you can just play around with the factors until you discover it:

Two Numbers                                             Thinking ...                 Highest

Common Factor   Example Simplified

Fraction

9 and 12     3 × 3 = 9 and 3 × 4 = 12               3                                    3

9/12 = ¾    which dived both numbers

Highest Common Factor of 12 and 30

1. Find all the Factors of each number,

2. Circle the Common factors,

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

Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30

3. Choose the Highest of those

Factor 6 is the Highest of 1,2,3,6

A number can have many factors:

Example: The common factors of 15, 30 and 105

Factors of 15 are 1, 3, 5, and 15

Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30

Factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105

The factors that are common to all three numbers are 1, 3, 5 and 15

In other words, the common factors of 15, 30 and 105 are 1, 3, 5 and 15

What is the "Highest Common Factor”?

It is simply the largest of the common factors.

In our previous example, the largest of the common factors is 15, so the Highest Common Factor of 15, 30 and 105 is 15