gcf help (greatest comon factor)?

Question

Explain how to find the greatest common factor of two or more numbers.

Find the common factors of 21 and 45 by using factor trees

Name two numbers whoose GCF is 15

Find the GCF of each set pf numbers or monomials.

12,8

12x,40x squared

Find the GCF of each set of numbers or monomials.

16,56

6,8,12

20,21,25

Answer 1

The greatest common factor is the greatest number that divides into the other numbers evenly.

For 21, 45
21 = 3 * 7
45 = 3 * 3 * 5
The most they have in common is 3 (once)
So the answer is 3.

Similarly for the other parts:

Two numbers whose GCF is 15 could be 2*15 and 3*15 which would be 30 and 45.

12,8
12 = 2 * 2 * 3
8 = 2 * 2 * 2
So the GCF is 2 * 2 = 4

12x, 40x
12x = 2 * 2 * 3 * x
40x^2 = 2 * 2 * 2 * 5 * x * x
So the GCF is 2 * 2 * x = 4x

16, 56
16 = 2 * 2 * 2 * 2
56 = 2 * 2 * 2 * 7
The GCF is 2 * 2 * 2 = 8

6, 8, 12
6 = 2 * 3
8 = 2 * 2 * 2
12 = 2 * 2 * 3
the GCF is 2

20, 21, 25
20 = 2 * 2 * 5
21 = 3 * 7
25 = 5 * 5
The GCF is 1. They have no common factors.

Answer 2

To find the GCF, use Euclid's algorithm.

It applies to two numbers. To apply it to more than two, first apply it to two of them, then apply it to their GCF and a third number to get the GCF of those 3, then apply it to their GCF and a fourth number to get the GCF of those 4, and so on.

For example, to find the GCF of 36 and 64:

64/36 = 1, with remainder 28
36/28 = 1, with remainder 8
28/8 = 3, with remainder 4
8/4 = 2, with remainder 0. Stop. The GCF is 4.
Note that each step uses the smaller number from the previous step, and divides that by the remainder from the previous step. Keep going until you have a 0 remainder.

Factor trees: For each number, 21 or 45, divide the number into two factors. If any factor is not prime, divide it again, and keep going, until all factors are prime. Once you do this with both numbers, you can examine the "leaves" of the two trees and find common factors.

Answer 3

6