Greatest Common Factor

This is the greatest Common Factor between 2 or more numbers.

To find the GCF of numbers (let’s say 8 & 10):

  • 1) List all the prime factors for each number.
  • 2) Find the factors which are common in both the numbers.
  • 3) Multiply all the common factors from step # 2 to get Greatest CommonFactor or GCF.

Note:

  •  GCF is also called Greatest Common Divisor (GCD).
  •  GCF is also called sometimes Highest Common Factor (HCF).
  •  GCF is closely related with the LCM.

Simple Examples


Example 1:
 5

Explanation:

We will find the prime factors of each number.So that,
35 = 5×7
15 = 5×3
There is only one factor in common. That is 5. Therefore, GCF = 5

Example 2:
 5

Explanation:

We will find the prime factors of each number.So that,
35 = 5×7
10 = 5×2
There is only one factor in common. That is 5. Therefore, GCF = 5

Example 3:
 25

Explanation:

We will find the prime factors of each number.So that,
25 = 5 × 5
125 = 5 × 5 x 5
There are two common factor 5 and 5 and their multiplication is 25. Therefore, GCF = 25

Example 4:
 15

Explanation:

We will find the prime factors of each number.So that,
15 = 5 × 3
30 = 5 × 3 x 2
There are two common factor 5 and 3 and their multiplication is 15. Therefore, GCF = 15

Example 5:
 30

Explanation:

We will find the prime factors of each number.
So that,
240 = 24 × 10
= 3 × 8 × 10
= 3 × 2 × 2 × 2 × 2 × 5
150 = 15 × 10
= 5 × 3 × 2 × 5

Identify the common factors in both numbers. They are 3, 2 and 5. To find GCF of the numbers multiply the common factors between two numbers. GCF=3×2×5= 30

Example 6:
 25

Explanation:

If you have given three numbers 125, 450 and 526.
To find GCF of the numbers we will find the prime factors of each number.
So that,
125 = 25 × 5
= 5 × 5 × 5
450 = 45 × 10
= 5 × 9 × 2 × 5
= 5 × 3 × 3 × 2 × 5
525 = 5 × 5 × 21
5 × 5 × 7 × 3
I can see that the all numbers have a factor of 5, 5 in common. Multiply these numbers 5 × 5 = 25.
So that, GCF = 25


GCF of Polynomials


Polynomials are Terms containing multiples of various variable( x, y, z etc.)

 Steps to get GCF of 2 polynomials

To find the GCF of all the terms in the polynomial

  •  Find which variables appear in every term of the polynomial.
  •  Factor the coefficient of each term in the polynomial using prime factorization.

Simple Example:

 GCF of x2 and x3 is x2.
 GCF of 2y2& 4y is 2y.


Advance Example:

 2xy

Explanation:

In the polynomial 6x3y2, 8x2y and 4xyz for example, x and y appear in all three terms, so they are both part of the GCF. Z appears in only one term, so do not include it in the GCF.

Use each variable with the lowest exponent it has in the polynomial for the GCF. In 6x3y2, 8x2y and 4xyz. 1 is the lowest exponent with x, The lowest exponent of y is 1. This means xy is part of the GCF for this polynomial.

The coefficients of 6x3y2, 8x2y and 4xyz are 6, 8, and 4. The prime factorizations are 2×3, 2× 2 × 2 and 2× 2, respectively. Since only 2 is common factor in all, it is the GCF.
Write the GCF of the coefficients before the GCF of the variables to get the complete GCF for the polynomial.

Therefore, GCF of 6x3y2, 8x2y and 4xyz is 2xy.


 Points to Remember

Quick Tips

  •  For 2 numbers LCM is bigger than GCF.
  •  GCF is always smaller or at most equal to the smallest numbers.

Think

  •  For two given numbers a, b; GCM will be bigger than x&y or smaller than x&y.
  •  Can GCF be ever equal to one of the number given out of 2 numbers x & y? If yes,please explain the scenario?





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