Always, sometimes, or never

A good way to test your grip on divisibility: take a statement and decide whether it is always true, sometimes true, or never true. Each verdict needs a different kind of evidence.

Statement 1: if 8 divides two numbers separately, it divides their sum. Write them 8a and 8b. Then 8a + 8b = 8(a + b): a multiple of 8, no exceptions. Always true. (Subtraction too: 8a − 8b = 8(a − b).)

Law: if a divides M and a divides N, then a divides M + N and M − N.

Statement 2 turns it round: if a number is divisible by 8, must any two numbers that add up to it also be divisible by 8? 72 = 48 + 24, and both parts are multiples of 8. But 72 = 50 + 22, and neither is. Sometimes true, and one counterexample was all it took.

Statement 3: if a number is divisible by 7, are all its multiples? Write the number 7j; its multiples are (7j) × m = 7jm, which plainly has a factor of 7. Always true. (21, 42, 63, 84 …)

Law: if A is divisible by k, then all multiples of A are divisible by k.

Statement 4: if a number is divisible by 12, is it divisible by every factor of 12? Write it 12m = 2 × 6 × m = 3 × 4 × m. Every factor of 12 is sitting inside. Always true. (48 is divisible by 1, 2, 3, 4, 6 and 12.)

Law: if A is divisible by k, then A is divisible by all the factors of k.

Statement 5: if a number is divisible by 7, is it divisible by every multiple of 7? 42 is divisible by 7 and by 14, but not by 28. Sometimes true. The exact rule: 7k is divisible by 7m exactly when m is a factor of k.

Then the fourth law, about two divisors at once. If A is divisible by k and by m, then A is divisible by LCM(k, m), because A's prime factorisation must contain the prime factorisation of that LCM. So divisible by 9 and by 4 forces divisible by 36; but divisible by 6 and by 4 does not force divisible by 24, because LCM(6, 4) is 12, not 24.

Finally a never. 'Adding an odd number to an even number gives a multiple of 6.' Multiples of 6 are all even, but odd + even is always odd. In algebra: if 2n + (2m + 1) = 6j, then 2(n + m) = 6j − 1, which says an even number equals an odd one. Never true.