# Greatest common factor

*Common factor of some numbers.*

Greatest common factor (GCF). Finding GCF.

Greatest common factor (GCF). Finding GCF.

*Common factor* of some numbers – a number, which is a factor of each of them. For example, numbers 36, 60, 42 have

common factors 2 and 3 . Among all common factors there is always the greatest one, in our case this is 6. This number

is called a ** greatest common factor **(GCF).

To find a ** greatest common factor **(GCF) of some numbers it is necessary:

1) to express each of the numbers as a product of its *prime factors*, for example:

2 ·

2 ·

3 ·

3 ·

5 ,

2) to write *powers of all prime factors* in the factorization as:

2 ·

2 ·

3 ·

3 ·

5 = 2

^{3}·

3

^{2}·

5

^{1},

3) to write out all *common factors* in these factorizations;

4) to take *the least power *of each of them, meeting in the all factorizations;

5) to multiply these powers.

E x a m p l e . Find GCF for numbers: 168, 180 and 3024.

S o l u t i o n . 168 = 2

· 2

·

2 ·

3 ·

7 = 2^{3} ·

3^{1} ·

7^{1} ,

180 = 2

· 2

·

3 ·

3 ·

5 = 2^{2} ·

3^{2} ·

5^{1} ,

3024 = 2

· 2

·

2 ·

2 ·

3 ·

3 ·

3 ·

7 = 2^{4} ·

3^{3} ·

7^{1}** **.

Write out the least powers of the common factors 2 and 3 and multiply them:

^{2}·

3

^{1}= 12 .