The idea behind the proof of the rules

The divisibility rules are not magic tricks: they all arise from the study of remainders, that is, from congruences.

Remark — The idea behind the proof

The rule for $3$ (and for $9$) follows from $10\equiv 1\pmod 3$: every power of $10$ is $\equiv 1$, so a number $\overline{a_k\ldots a_1 a_0}=\sum a_i\,10^i\equiv\sum a_i\pmod 3$, that is, it has the same remainder as the sum of its digits. The rule for $11$ follows instead from $10\equiv -1\pmod{11}$: we have $10^i\equiv(-1)^i$, from which the alternating difference of the digits.

Congruences will be taken up systematically later on, in the study of number theory linked to RSA cryptography.

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Topics: Numbers and operations
Concepts: Congruence · Divisibility rules · Divisibility
Skills: Prove