Given that a,b are integers such that $2a+3b=6$, I'm trying to prove:
i) $\frac{a}{3},\frac{b}{2}$ are integers
ii) gcd($\frac{a}{3},\frac{b}{2}$)=1
I know that, the solution of this linear diophantine equation is $a = -6 + 3n$ and $b = 6 - 2n$ where n is an integer.
From the solutions I can write:
$$\frac{a}{3}=-2+n\\\frac{b}{2}=3-n$$
Since n is an integer i) is proven. However, I have no idea how to move on with the second proof
We have that $$ 1=\frac{2a+3b}6=\frac a3+\frac b2$$ is a multiple of $\gcd(\frac a3,\frac b2)$.