elementary number theory - How to prove or disprove that $gcd(ab, c) = gcd(a, b) times gcd(b, c)$?

I'm new to proofs, and am trying to solve this problem from William J. Gilbert's "An Introduction To Mathematical Thinking: Algebra and Number Systems". Specifically, this is from Problem Set 2 Question 74. It asks:

How to prove or disprove that $\gcd(ab, c) = \gcd(a, b) \times \gcd(b, c)$?

What I've tried is to use the proposition that $\gcd(a, b) = ax + by$ to rewrite the whole equality, but I can't manage to equate the two statements.

Any help would be appreciated.

Answer

Notice if $a = b = c = 3$, then

$$ \gcd(ab,c) = \gcd(9,3) = 3 $$

while

$$ gcd(a,b) \times gcd(b,c) = gcd(3,3) \times gcd(3,3) = 3 \times 3 = 9 $$

$$ \therefore gcd(ab,c) \neq gcd(a,b)\times gcd(b,c) $$