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

^{ 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 .