Level 5: Clock arithmetic

Number theory · level 5 of 8 · rough age 12-13 (a guide for parents; anyone can start anywhere)

In level 4 the remainder was the engine of Euclid's algorithm. In this level the remainder becomes a number in its own right: every whole number is placed on a clock, and you add, subtract and multiply on the clock. You will prove the one rule that makes this work ("reduce whenever you like"), use it to find the last digits of numbers with hundreds of digits, see at last why the divisibility tests from level 3 work, and find out how a single extra digit catches typing mistakes in book and card numbers.

Lesson	You will be able to
Remainders as a new kind of number	Write \(a = qb + r\) with \(0 \le r < b\), find remainders of negative numbers, and prove facts by checking each remainder family
Clocks of any size	Add, subtract and multiply on any clock, build the tables mod 5 and mod 6, and spot which clocks let you divide
Congruence	Use \(a \equiv b \pmod n\), prove the "reduce whenever you like" rule, and know which moves are not allowed
Last digits and big powers	Find the last digit (or last two digits) of huge powers using cycles, and explain why the cycles must exist
The divisibility tests explained	Explain the tests for 2, 3, 4, 5, 8, 9 and 11, test for 7, 11 and 13 at once, and check multiplications by casting out nines
Check digits	Find and test ISBN-13 and Luhn check digits, and prove which mistakes they catch and which they miss

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