A ten is a nine and one

About this video

Three friends win 471 sweets and want to share them equally. The last digit can't tell you whether that works, because the last-digit test only works for 2, 5 and 10. For 3 there is another test: add up the digits. The digit sum of 471 is 4 + 7 + 1 = 12, and 12 divides by 3, so 471 divides by 3 too, 157 sweets each. The test for 9 works the same way.

This short animation shows why the test works. Take ten dots and put nine of them aside: nine splits into three threes, and into one nine, with one dot left over. So every ten is a nine plus one. Now look at 42, which is four tens and two ones. Each of the four tens leaves one over after its nine, and the two ones are left over too, so everything left over is 4 + 2 = 6. That is exactly the digit sum. The nines are never any trouble, so only the leftovers decide, and the leftovers are the digit sum. 6 divides by 3, so 42 divides by 3. This video goes with the Tests for 3 and 9 lesson of Number Theory Level 1 on Knaph.

Transcript

1. 0:00Four hundred and seventy-one sweets, three friends. Fair shares?
2. 0:04Add up the digits. Four, seven and one make twelve.
3. 0:10Twelve divides by three, so four seventy-one does too.
4. 0:15Why? Put nine of the ten aside. One is left. A ten is a nine and one.
5. 0:22Forty-two is four tens and two ones. The leftovers make six.
6. 0:29Six divides by three, so forty-two does too.
7. 0:34The nines don't matter. Only the leftovers decide.