Chinese remainder theorem? In the 2014 AIME 1, number 8 says:
The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$.
I solved this problem using modular arithmetic and a little bit of logic (mainly the realization that if $N^2 - N$ is congruent to $0 \pmod{10000}$ then either $N$ is divisible by $2^4$ and $N-1$ is divisible by $5^4$ or vice versa.)
I saw a solution that used the Chinese remainder theorem, something I've never seen before. How does this theorem work, and how would it apply to this problem?

You could start with Wikipedia  A web search will turn up many references.  It looks like you were applying it without knowing it.  Here you are looking to solve $N \equiv 0 \pmod {2^4}, N\equiv -1 \pmod {5^4}$ or the other way around.  Because $2^4,5^4$ are relatively prime, CRT says there will be exactly one solution $\pmod {2^4\cdot 5^4}$  Note that without $a \neq 0$ the strings $0000$ and $0001$ also solve the problem.
