The test that alternates the signs

A bus board lists routes 91, 121 and 209. Which divide by 11?

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.

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.

Try the lab to finish this lesson.