Coprimality of 3a+2 and 5a+3

Statement

Prove that for every integer $a$ the numbers $3a+2$ and $5a+3$ are coprime, i.e. $\gcd(3a+2,\,5a+3) = 1$.

Solution

Let $d$ be a common divisor of $3a+2$ and $5a+3$. Then $d$ divides every integer linear combination of them. Consider $5\,(3a+2) - 3\,(5a+3) = 15a + 10 - 15a - 9 = 1.$ So $d \mid 1$, which forces $d = 1$ (up to sign). Hence the only positive common divisor is $1$ and the two numbers are coprime. $\blacksquare$