Obviously, we will not be looking here for a divisibility rule for zero: dividing by zero is simply forbidden — that is the absolute taboo. Nor will we be interested in dividing by one, because that, by contrast, is too easy and always possible.
The first serious divisibility rules are those for 2, 5 and 10. They have in common that they use only the units digit: if the units digit of a number n is 0, 2, 4, 6 or 8 then n is even (that is, divisible by 2), and conversely. If this same digit is 0 or 5, then it is a multiple of 5, and conversely. And so, n is a multiple of 10 if, and only if, its units digit is a 0. So much for the elementary rules.
Only slightly less simple is the case of divisibility by d = 2 *k, 5 k or 10 k: for these values of d, a number n is divisible by d if, and only if, its k last digits form a number divisible by d*.
The divisibility rules for 3 and 9 are famous too: n is a multiple of 3 (or of 9) if, and only if, the sum s of its digits is too. Applying this same rule to s again if necessary, and repeating as needed, we eventually get a number small enough to settle the question. In fact, the rule is even more precise: the remainder of the division of n by 3 (or by 9) is the same as the remainder of the division of s by 3 (or by 9).
The divisibility rule for 11, which will be proved later, is somewhat less well known: n is divisible by 11 if, and only if, the alternating sum s of its digits is divisible by 11. So, for n = 1,244,245, we have s = 1 − 2 + 4 − 4 + 2 − 4 + 5 = 2, so n is not a multiple of 11. (We can even say that the remainder of its Euclidean division by 11 is 2.)
