Suppose $a$ and $b$ are integers so that not both $a$ and $b$ are zeroes. Prove that not both $3a+2b$ and $5a+3b$ are zeroes

Question

The is from an introduction discrete mathematics course.

Suppose $a$ and $b$ are integers so that not both a and b are zeroes. Prove that not both $3a+2b$ and $5a+3b$ are zeroes and that $gcd(3a+2b, 5a+3b)$ = $gcd(a,b)$.

I understand how to prove that $gcd(3a+2b, 5a+3b)$ = $gcd(a,b)$ but not sure how to prove the initial part: "Suppose $a$ and $b$ are integers so that not both $a$ and $b$ are zeroes. Prove that not both $3a+2b$ and $5a+3b$ are zeroes."

How do I go about proving the initial statement?

Answer 1

Assume that both $3a+2b$ and $5a + 3b$ are equal to zero.

Then we get that $15a + 10b = 15a + 9b \implies b = 0$. Therefore we must have that $a \not = 0$. Hence we have that $3a + 2b = 3a \not = 0$. Therefore we obtain a contradiction, so at least one of $3a+2b$ and $5a+3b$ isn't zero.