Greatest common factor
Common factor of some numbers. 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:
360 = 2 ·
2 ·
2 ·
3 ·
3 ·
5 ,
2) to write powers of all prime factors in the factorization as:
360 = 2 ·
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:
GCF = 2^{2} ·
3^{1} = 12 .
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