Chinese Remainder Theorem not necessarily coprime moduli (part of proof) The usual Chinese Remainder Theorem requires that the moduli be pairwise relatively prime.  A modification states that a given system of linear congruences $x\equiv a_i\pmod{m_i}$ is solvable if and only if $a_i\equiv a_j\mod{(n_i,n_j)}$ for all $i\neq j$.
I've looked around and I've got it narrowed down to this:
The system consisting of the congruences $x\equiv a\pmod{p}$ and $x\equiv b\pmod{p^2}$ is solvable if and only if $a\equiv b\pmod{p}$.
What I can't seem to find is how $a\equiv b\pmod{p}\Rightarrow$ a solution exists.

If $a\equiv b\pmod{p}$, then $x=b$ is a solution :-)
More generally, if we're given a prime number $p$ and integers $a$ and $b$ with $a\equiv b\pmod{p}$, the solutions to the system
$$\begin{align}
x&\equiv a\pmod{p}\\
x&\equiv b\pmod{p^2}
\end{align}$$
are precisely
$$x\in\{b+kp^2\mid k\in\mathbb{Z}\}.$$
