Find the possible values gcd of a combination of two coprimes

Question

$a$ and $b$ are coprimes, so $\gcd(a, b) = 1$. Find the values of $\gcd(3a - b, 2a + b)$.

Tried to use Bezout's identity, but I am totally stuck. Any hint?

Answer 1

If $d$ divides $A$ and $B$, then $d\mid A+B$.

So in your case, a divisor of $3a-b$ and $2a+b$ divides

$$3a-b+2a+b=5a.$$

More generally, if $d$ divides $A$ and $B$, then $d\mid uA+vB$.

So you also have

$$d\mid 2(3a-b)-3(2a+b)=-5b.$$

So $d\mid 5$ since $a$ and $b$ are coprimes.

So the gcd you are looking for is $5$ or $1$.

Answer 2

If $d$ divides both, $d$ must divide $2(3a-b)-3(2a+b)=-5b$

and $3a+b+2a-b=5a$

So, $d$ must divide $(5b,5a)=5(b,a)$