coomon factors

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I don't understand common factors. can someone show me how to find the greatest common factor of 385 and 1365?

385 = 5 x 7 x 11
1365 = 3 x 5 x 7 x 13

which factors are found in both?
5 x 7
so 35 is the HCF, (it's like taking the intersection of the elements of two sets)

There are two methods. The first one is the brute force method where you just factor each of the numbers in prome factors and see which prime factors they have in common.

385 = 5 * 7 * 11

1365 = 3 * 5 * 7 * 13

So, the greatest common factor (a.k.a. GCD or gretest common divisor) is 5 * 7 = 35

But there is another (much faster for large numbers) method that doesn't involve factorization. Since the GCD of two nu,mbers doesn't change when you replace one of the numbers b a linear combination of the two numbers you can subtract from the largests of the two numbers a multiple times the smallest number so that it becomes smaller than the smallest if the two. If you repeat this process with now the new numbers, then the numbers will become smaller and smaller until you end up with a trivial case.

GCD(385, 1365) =

GCD(1365 - 3*385, 385) =

GCD(210, 385) =

GCD(385-210,210) =

GCD(175,210) =

GCD(35,175) =

GCD(0,35) = 35

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