A friend has sent this question to me. I have it solved with linear algebra (and some hand waving). Can you find a shorter way to the answer? (It's not a homework, not mine anyway. My homework times long gone.)
A gardener collected 17 apples. He finds that each time he
removes an apple from his harvest, he can share the remaining fruit
in two piles of equal weight, each containing 8 apples. Show that
all apples are the same weight.