The test that alternates the signs
A bus board lists routes 91, 121 and 209. Which divide by 11?
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- 209 has digits 2, 0, 9.
- Start at the ones digit. Give it a plus: + 9.
- Give the next digits alternating signs: − 0, + 2.
- 9 − 0 + 2 = 11. A multiple of 11, so 209 is too.
The rule
121 and 209 divide by 11 with nothing left over. 91 does not.
A place is a position in a number: ones, tens, hundreds. Its worth is 1, 10, 100, 1,000 and so on. Compare each worth with a multiple of 11. 1 is one more than 0. 10 is one less than 11. 100 is one more than 99. 1,000 is one less than 1,001, which is 11 × 91. So the places go one more, one less, one more, one less. The signs go plus, minus, plus, minus.
Start at the ones digit with a plus. Give the next digit a minus, then a plus, and keep alternating. The total is the alternating sum.
Rule: Alternate plus and minus from the ones digit. Zero or a multiple of 11 passes.
| Number | Digit sum | Multiple of 3? | Alternating sum | Multiple of 11? |
|---|---|---|---|---|
| 123 | 6 | yes | 2 | no |
| 209 | 11 | no | 11 | yes |
| 456 | 15 | yes | 5 | no |
| 253 | 10 | no | 0 | yes |
The alternating sum also gives the remainder, the number left over. Add or subtract 11 until the result is from 0 to 10. For 456 it is 6 − 5 + 4 = 5. And 456 ÷ 11 leaves remainder 5.