Trick of Mind

If p and q are distinct primes, show that p^q + q^p = p + q (mod pq)

Prove that 22225555 + 55552222 is divisible by 7.
I know, it’s boring
Obviously, no calculators or computers are allowed.

I have four positive numbers, a, b, c and d (not necessarily integers).
Obviously, there are six ways to multiply pairs of them, yielding the products ab, ac, ad, bc, bd and cd.
I tell you what five of these products are (but not which product is what):
2, 3, 4, 5 and 6
What’s the [...]

Bill and John take turns rolling a fair die. The winner is the first to roll a 6. Bill rolls first. If John wins, what is the probability he did so on his second roll?