Program of Lessons

# 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 = 23  ·  32  ·  51 ,

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 = 23  · 31  · 71 ,

180 = 2 · 2 · 3 · 3 · 5 = 22  · 32  · 51 ,

3024 = 2 · 2 · 2 · 2 · 3 · 3 · 3 · 7 = 24  · 33  · 71 .

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

GCF = 22  · 31  = 12 .

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