Divisibility tests for 10, 5, 2, 4 and 8: why only the last digits matter

0:00Here's a number. Eight thousand five hundred and sixty. And here are nine candidates. Two, three, four, five, six, seven, eight, nine and ten. Which of them divide it exactly? You could do nine divisions. Or read the end of the number and know most of the answers at once. Have a guess now. How many of the nine are factors? Hold on to your number. By the end you can check it in seconds.

0:31Start with the easiest. Ten. Count in tens and watch the endings. Ten, twenty, thirty, forty, fifty, sixty. Every single one ends in a zero. And nothing else does. So a number is divisible by ten exactly when its last digit is zero. Test it. Is one hundred and twenty-five a multiple of ten? It ends in five. No. And our big number ends in zero. So ten divides it.

1:02One down. Now five. Same method. Count in fives and watch the ending. Five, ten, fifteen, twenty, twenty-five, thirty, thirty-five, forty. The endings go five, zero, five, zero, over and over. So five divides a number when it ends in five or zero. Two endings instead of one. What's the largest number below three hundred and ninety-nine that five divides? Count back. Ninety-eight, no. Ninety-seven, no. Ninety-six, no. Ninety-five, yes.

1:39And our big number ends in zero, so five divides it too. Now two. Count in twos. Two, four, six, eight, ten, twelve, fourteen, sixteen, eighteen, twenty. The endings cycle through two, four, six, eight, zero. Five possible endings. So a number is even exactly when its last digit is one of those five. Six hundred and eighty-two ends in two, so it's even. Our big number ends in zero, so it's even too.

2:14Three of our nine settled, and we haven't divided anything. The last digit is doing well. Let's push our luck and try four. Twelve is divisible by four. Twenty-two isn't. And they both end in two. Fourteen isn't divisible by four. Twenty-four is. Both end in four. Sixteen is. Twenty-six isn't. Both end in six. Eighteen isn't. Twenty-eight is. Both end in eight. Four times over, two numbers with the same last digit disagree about four.

2:52So the last digit simply cannot decide this one. It doesn't carry enough information. That's worth sitting with. It's where most people stop and start memorising. We don't need a new idea. We need a wider window. One digit wasn't enough. Let's look at two. Here are the multiples of four between three thirty and three forty. Three hundred and thirty-two, three hundred and thirty-six, three hundred and forty. Now the same window a thousand higher. Seventeen thirty to seventeen forty.

3:30Seventeen thirty-two, thirty-six, and forty. And once more, two thousand and thirty to two thousand and forty. Thirty-two, thirty-six, forty. Look at the last two digits in all three windows. Thirty-two, thirty-six, forty. Identical, every time. The thousands changed completely, and the answer didn't care. So the test for four. Ignore all but the last two digits. If those are divisible by four, so is the number. Our big number ends in six zero. Sixty over four is fifteen. So four divides it.

4:12Now eight, and you can probably guess what's coming. Multiples of eight between one twenty and one forty. One twenty, one twenty-eight, one thirty-six. The same window a thousand up. Eleven twenty, eleven twenty-eight, eleven thirty-six. And three thousand up. Thirty-one twenty, thirty-one twenty-eight, thirty-one thirty-six. The last three digits are the same in all three. One twenty, one twenty-eight, one thirty-six. So for eight, the window is three digits wide.

4:49Our big number. Last three digits, five sixty. Over eight, that's seventy. So eight divides it. Why does widening the window work? Because this isn't five tricks. It's one fact, used three times. Look at how we write eight thousand five hundred and sixty. Everything left of the last digit is a whole number of tens. Eight hundred and fifty-six of them. So if your divisor goes cleanly into ten, all of that vanishes, leaving the final digit.

5:24Two, five and ten all go into ten. That's why the last digit settles them. Now move the wall one place left. Everything beyond the last two digits is a whole number of hundreds. Four goes into a hundred exactly. So for four, everything above the last two digits vanishes. And one more. Everything left of the last three digits is a whole number of thousands. Eight goes into a thousand exactly. So for eight, the last three digits are all that matter.

6:00That's the whole section. Not five rules. One idea, with the wall in three different places. Which immediately tells us where this has to stop. It only works if your divisor goes exactly into ten, a hundred, or a thousand. So try three. Ten over three leaves one. A hundred leaves one. A thousand leaves one. Three doesn't go cleanly into any of them. And neither do six, seven or nine.

6:32So no amount of staring at the tail of a number will ever settle those four. That's not laziness on anyone's part. It's the method genuinely running out. Tests for three, six, seven and nine do exist, but they work a different way, and wait for a later year. So three, six, seven and nine are the ones we still can't shortcut. And none of them divide our number. Five of the nine were factors. Two, four, five, eight and ten.

7:05Now a lovely puzzle. Does fourteen thousand five hundred and sixty pass all five tests? Two, four, five, eight and ten. A student checks only two of them, and then announces all five. Which two would you check? Here's one answer. Check eight and check ten. If eight divides it, four does too, since four goes into eight. And two into four. So a single check on eight carries eight, four and two.

7:36And if ten divides it, five does too, since five goes into ten. So eight and ten between them carry all five. Two checks, five answers. And it ends in five six zero. Divisible by eight, and ends in zero. So all five pass. One more habit worth building, and it is easy to walk past. There are two claims here, not one, and they point in opposite directions. First claim. If the last two digits are divisible by four, then the whole number is.

8:12Second claim, the other way round. If the number is divisible by four, its last two digits are. They sound like the same sentence, and they aren't. For this test both directions happen to be true, which is what makes it usable. But do not assume that always happens. Here's one that fails. If a number is divisible by eight, is it divisible by four? Yes, always. Turn it round. If a number is divisible by four, is it divisible by eight? Twelve. Twenty. Twenty-eight.

8:49All divisible by four, and none of them by eight. That direction is false. Same two numbers, two different questions, two different answers. Always check both ways. Let's spend these tests on something real. Leap years. A year is a leap year when four divides it, unless a hundred does and four hundred doesn't. How many leap years from twenty twenty-four to twenty ninety-nine? The multiples of four run from twenty twenty-four to twenty ninety-six. Nineteen of them.

9:25And the exception never bites: there's no century year in that range. Now palindromes. Numbers that read the same backwards, like two one one two. What's the smallest four-digit palindrome divisible by four? And the largest? Use the test. Only the last two digits matter, and in a palindrome those are the first two, reversed. The smallest is two one one two, and the largest is eight eight eight eight.

9:56And one last puzzle to finish on. Ten thousand is two times two times two times two, times five times five times five times five. Split it into two factors so that neither one ends in a zero. A number ends in zero only if it has both a two and a five. So give all the twos to one side and all the fives to the other. Sixteen and six hundred and twenty-five. Neither ends in zero, and they multiply to ten thousand.

10:31Here's my question to leave you with. We found tests for two, four, five, eight and ten by looking at the end of a number. What would a test for one hundred and twenty-five look like, and how wide its window? Tell me in the comments.