Divisibility Tests

A divisibility test tells you whether a number divides exactly without doing the division. The easiest three look only at the last digit. A number is divisible by two if its last digit is even, by five if its last digit is zero or five, and by ten if its last digit is zero. The reason is that every number is some tens plus its last digit, and two, five and ten all divide any number of tens — so only the last digit can spoil it. Four thousand seven hundred and twenty ends in zero, so it passes all three.

Four and eight work the same way, with more digits. A hundred is four times twenty-five, so any number of hundreds divides by four; only the last two digits can spoil it. Five thousand three hundred and sixteen: sixteen divides by four, so the whole number does. A thousand is eight times one hundred and twenty-five, so only the last three digits matter for eight. Two thousand three hundred and sixteen: three hundred and sixteen divided by eight is thirty-nine and a half, so it is not divisible by eight — though it is by four, since sixteen is. Eight implies four, never the reverse.

For three and nine, add the digits. If the digit sum divides by three, so does the number; if it divides by nine, so does the number. Four thousand seven hundred and twenty-five: four plus seven plus two plus five is eighteen, which nine divides, so the number divides by nine and by three. Five hundred and sixty-one has digit sum twelve: divisible by three but not nine. The reason is that ten, a hundred and a thousand are each one more than a multiple of nine, so when you split a number into its place-value parts, every part except the digit sum is a multiple of nine. Whatever is left over comes entirely from the digits.

There is no separate test for six; you use two you already have. Six is two times three, and two and three share no factor, so a number divides by six exactly when it passes both the test for two and the test for three. Two hundred and thirty-four is even and its digits add to nine — divisible by six. Three hundred and forty-five passes for three but is odd, so not six. The same combining works for twelve, three and four, and fifteen, three and five. It does not work for eight as two and four, because two and four overlap: twelve passes both and is not divisible by eight.

Two places the tests earn their keep. The missing-digit puzzle: five, blank, four is divisible by three — what can the blank be? The digits you have add to nine, so the blank must add to a multiple of three: zero, three, six or nine. For divisibility by nine the answer is usually a single digit: four two blank seven eight has digit sum twenty-one plus the blank, and only six makes twenty-seven. And testing for primes: two, three and five are settled at a glance, leaving only seven to actually divide. Quiz and exercises now.