Divisibility Rules

You can often tell whether a number divides by 2, 5 or 10 from its last digit alone. A number divides by 2 when it ends in 0, 2, 4, 6 or 8 (it is even), by 5 when it ends in 0 or 5, and by 10 when it ends in 0. The reason: every number is some tens plus its last digit, and tens always divide by 2, 5 and 10, so only the last digit can leave anything over.

For 3 and 9, add the digits instead. 471 has the digit sum 4 + 7 + 1, which is 12. 12 divides by 3, so 471 does too: 471 / 3 is 157. 12 does not divide by 9, so 471 does not.

Why the digit sum? 10 is one more than 9, and 100 is one more than 99, so 471 is a pile of nines plus 4 + 7 + 1 left over. The nines divide by 3 and by 9, so whatever the digit sum does, the whole number does too.