Ladder
Modular arithmetic — the ladder
2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
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Numbers that wrap
A clock does arithmetic. It has finitely many numbers, addition never leaves it, and multiplication behaves entirely differently depending on one property of the size of the dial.
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Two dials at once
Watch one number on two clocks with different faces. If the faces share no factor, every pair of readings occurs exactly once — so two remainders name a number, and a hard calculation can be split into two easy ones.