The positive integer n is not divisible by 7. The remainder when n^2 is divided by 7 and the remainder when n is divided by 7 are each equal to k. What is k?

You have to think pretty nimbly about remainders here. Start by thinking of some perfect squares you know: 64, 81, 100, etc. For each of those, what is the remainder when it’s divided by 7? 64/7 has a remainder of 1; 81/7 has a remainder of 4, and 100/7 has a remainder of 2.

Of those, what are the remainders when their square roots are divided by 7? 8/7 has a remainder of 1, and with that, we have a winner. 8/7 has a remainder of 1, and 64/7 has a remainder of 1. From that we know that k = 1.

This is weirdly true more than once: it’s also the case that 15 and 15^2 both give a remainder of 1 when divided by 7; same is true for 22 and 22^2.  Who knew?!