Problem. We say that the $n$-digit number $x$ is automorphic iff $x^2\equiv x \mod(10^n)$. Prove that if $x$ is $n$-digit automorphic number then $(3x^2-2x^3)\mod(10^{2n})$ is $2n$-digit automorphic number. Hint: use Chinese reminder theorem to find the necessary and sufficient condition for number to be automorphic.
So from Chinese reminder theorem we have that $x$ is automorphic iff: $ \begin{cases} x(x-1)\equiv 0 \mod(2^n)\\ x(x-1)\equiv 0 \mod(5^n)\end{cases} $ which gives us four systems of equations: $ \begin{cases} x\equiv 0 \mod(2^n)\\ x\equiv 0 \mod(5^n)\end{cases} $ $ \begin{cases} x\equiv 1 \mod(2^n)\\ x\equiv 1 \mod(5^n)\end{cases} $ $ \begin{cases} x\equiv 0 \mod(2^n)\\ x\equiv 1 \mod(5^n)\end{cases} $ $ \begin{cases} x\equiv 1 \mod(2^n)\\ x\equiv 0 \mod(5^n)\end{cases} $
and it's easy to check that thesis is true for the first two cases, just by simple operations. But how to check the last two?