Find The Mystery Number!

We want to compute $x \mod 66$ and because $66 = 6 \times 11$ , whatever this value is, it is uniquely determined (via the Chinese Remainder Theorem) by $x \mod 11$ and $x \mod 6$ .

First, we are already given that $x \equiv 7 \mod 11$ . And notice that via the (reverse) Chinese Remainder Theorem, $x \equiv 10 \mod 12 \implies x \equiv 10 \mod 6$ . We can simplify this like $x \equiv 4 \mod 6$ . We now have all we need. We just need to stitch up these congruences with the Chinese Remainder Theorem, which I will do in an informal way for the sake of clarity and brevity:

Let's suppose that $x = 6a + 11b$ . Then $6a \equiv 7 \mod 11 \implies a = 3$ . And we also have that $11b \equiv 4 \mod 6 \implies 5b \equiv 4 \mod 6 \implies b = 2$ . Thus $x = 6 \times 3 + 11 \times 2 = 18 + 22 = 40$ .