If A is a factor of BC , and GCD(A,B)=1 , then A is a factor of C.

Question

Hey! I just started studying for the GMAT and my math is very rusty. the problem is found in the GMAT Club Math book:

Can you explain what the significance of (GCD)(A,B)=1 means in this statement? I can factor out BC and A, but can't wrap my head around what the GCD(A,B) means

Bunuel · Answer

If a is a factor of bc, and gcd(a,b)=1, then a is a factor of c.

Say  a=2 ,  b=3  ( gcd(a,b)=gcd(2,3)=1 ), and  c=4 .

a=2  IS a factor of  bc=12 , and  a=2  IS a factor of  c .

OR: if  a  is a factor of  bc  and NOT a factor of  b , then it must be a factor of  c .

Hope it's clear.

cpowers · Answer

I still do not understand the significance of GCD in this equation. What is the purpose of this piece to identify A is a factor of C?

TestPrepUnlimited · Answer

Hi  cpowers  , the purpose is that we can make a clear conclusion A was "taken out of" C instead of B, and that C is a multiple of A. Let's say we have  5x = 13y  and both x and y are integers.

If we look at the equation this way as  x = \frac{13y}{5} , we can first obverse  \frac{13y}{5}  has to be an integer since x is an integer. We can see the 5 cannot be taken out of 13, there it must have been taken out of y. Then we can conclude y is a multiple of 5. Doesn't that seem useful?

Similarly, we can conclude x is a multiple of 13. Now back to the GCD = 1, this was to conclude that 5 cannot be factored from 13, so it must have been taken out of y.

Hopefully this helps!

Pragya9s · Answer

I was stuck on the same part.

This is what I found out - The Euclidean Algorithm Recall that the Greatest Common Divisor (GCD) of two integers A and B is the  largest integer that divides both A and B .
The  Euclidean Algorithm  is a technique for quickly finding the  GCD  of two integers.

karu26 · Answer

Given GCD(A, B) = 1, A and B are coprime, meaning they have no common prime factors. Since A is a factor of BC, and A shares no factors with B, all prime factors of A must come from C. Thus, A must divide C. Therefore, if GCD(A, B) = 1 and A | BC, then A | C.

If A is a factor of BC , and GCD(A,B)=1 , then A is a factor of C.

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If A is a factor of BC , and GCD(A,B)=1 , then A is a factor of C.

Prep Toolkit

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