SOLUTION: Hello! I have problems proving that if {{{GCD (a, b) = 1}}} then {{{GCD (a * c, b) = GCD (c, b)}}} Can someone help me on this, please? (I hope this is the correct topic cat

Algebra ->  Algebra  -> Divisibility and Prime Numbers -> SOLUTION: Hello! I have problems proving that if {{{GCD (a, b) = 1}}} then {{{GCD (a * c, b) = GCD (c, b)}}} Can someone help me on this, please? (I hope this is the correct topic cat      Log On

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Question 54308: Hello!
I have problems proving that if GCD+%28a%2C+b%29+=+1 then GCD+%28a+%2A+c%2C+b%29+=+GCD+%28c%2C+b%29
Can someone help me on this, please?
(I hope this is the correct topic category.)
Answer by aaaaaaaa(138) About Me  (Show Source):
You can put this solution on YOUR website!
I'm not sure if this explanation counts as a mathematical proof, but it should get you going:
if GCD%28a%2Ab%29+=+1 then a and b have no common factors at all (that's what causes the result to be 1).
Therefore, when you do a%2Ac you are adding all the factors of c to a. As an example:
6+=+2%2A3
25+=+5%2A5
6%2A25+=+2%2A3%2A5%2A5
So, if a%2Ac and b have a common factor, this factor is present in c, and GCD%28a%2Ac%2C+b%29+=+GCD%28c%2C+b%29