Prove that the system of congruences has a solution I'm doing the following exercise and I don't know how to prove it.
Suppose $a,b \in \mathbb{Z}$, and $d=\gcd(a,b)$.
I have to prove that if $x\equiv y \bmod d$, then the system 
$$X\equiv x \mod a$$
$$X\equiv y \mod b$$
has a solution.
I don't even how to start...
Thanks for your help :)


Bezout's Lemma $a,b$ are relatively prime if and only if there exists an $x,y$ such that $ax+by=1$.

Another way to phrase this is that $ax=1\pmod{b}$ and $by=1\pmod{a}$. See if you can use this statement to solve your problem.
