Divisibility rules

Hello, and welcome to Myriora Maths! Can one thousand, two hundred and thirty-six be split exactly into threes? You don't need to divide to find out. There are quick tests, called divisibility rules. Let's learn them, and see why they work!

Some rules just look at the last digit. A number divides by two if its last digit is even. It divides by five if it ends in five or zero. And it divides by ten if it ends in zero. Why only the last digit? Every ten already divides by two, by five, and by ten. So only the ones digit can spoil it. One thousand, two hundred and thirty-six ends in six. So it divides by two, but not by five or ten.

For four, look at the last two digits. A hundred divides by four, so the hundreds and thousands don't matter. This number ends in thirty-six, and thirty-six divided by four is nine. So it divides by four! For eight, look at the last three digits, because a thousand divides by eight. Two hundred and thirty-six divided by eight is twenty-nine and a half, so it doesn't divide by eight.

Now the clever ones. For three, add up the digits. One plus two plus three plus six is twelve. Twelve divides by three, so one thousand, two hundred and thirty-six does too! For nine, the digit sum must divide by nine. Twelve doesn't, so it doesn't divide by nine.

Why does adding the digits work? Split the number up. A thousand is nine hundred and ninety-nine, plus one. A hundred is ninety-nine, plus one. Ten is nine, plus one. So the number is a pile of nines, plus the digits added up. The pile of nines always divides by nine, and by three. So only the digit sum matters!

Six is two times three. So a number divides by six if it passes both tests. One thousand, two hundred and thirty-six is even, and its digit sum is twelve. So it divides by six: it's two hundred and six sixes!

Eleven has a fun rule. Start from the right, and take away and add the digits in turn. For two thousand, seven hundred and twenty-eight: eight, take away two, add seven, take away two. That's eleven. If the answer divides by eleven, or is zero, the number divides by eleven too. Two thousand, seven hundred and twenty-eight is two hundred and forty-eight elevens!

Here's a question that catches lots of people. Does a hundred and thirty divide by four? Some people add the digits: one plus three plus zero is four, so they say yes. But the digit sum test only works for three and nine! For four, look at the last two digits: thirty. Thirty divided by four is seven, remainder two, so the answer is no.

Your turn! Which of these numbers divides by nine? Pick the right card. You've got twenty seconds, or pause the video. Go!

Time's up! Add up the digits of each one. Four, three, two and one make ten. Five, three, eight and two make eighteen. Seven, one, zero and three make eleven. Only eighteen divides by nine, so the answer is five thousand, three hundred and eighty-two!

Quick review. For two, five and ten, look at the last digit. For four, look at the last two digits, and for eight, the last three. For three and nine, add up the digits. For six, pass the two test and the three test. And for eleven, take away and add the digits in turn. See you next time!