Question:
Suppose that a $30$-digit integer $N$ is composed of thirteen $7$s and seventeen $3$s. What is the remainder when $N$ is divided by $36$?

Answer:
By the Chinese Remainder Theorem, it suffices to find the remainders when $N$ is divided by $4$ and $9$. The last two digits of $N$ must be one of $33, 37, 73,$ or $77$; each of these leave a remainder of $1$ after division by $4$. By the divisibility property of $4$, it follows that $N \equiv 1 \pmod{4}$.

The sum of the digits of $N$ is equal to $13 \times 7 + 17 \times 3 = 142 = 15 \times 9 + 7$. This leaves a remainder of $7$ after division by $9$, so it follows that $N \equiv 7 \pmod{9}$.

By the Chinese Remainder Theorem and inspection, it follows that $N \equiv 25 \pmod{36}$ satisfies the two congruences, and so $N$ leaves a remainder of $\boxed{25}$ upon division by $36$.