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Greatest common factor
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:
to express each of the numbers as a product of its prime factors, for example:
360 = 2 · 2 · 2 · 3 · 3 · 5 ,
to write powers of all prime factors in the factorization as:
360 = 2 · 2 · 2 · 3 · 3 · 5 = 2
^{3}
· 3
^{2}
· 5
^{1}
,
to write out all common factors in these factorizations;
to take the least power of each of them, meeting in the all factorizations;
to multiply these powers.
Example:
Find GCF for numbers: 168, 180 and 3024.
Solution:
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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