Divisibility by 11, and other shortcuts

The test for 11 rests on the same idea as the test for 9 (look at what each place value leaves), but the leftovers alternate.

And so on, alternating for ever.

So some digits push you above a multiple of 11 and others pull you below it. The method has three steps:

• add the digits in the 'one more' places (units, hundreds, ten-thousands) to see how much you're over;
• add the digits in the 'one less' places (tens, thousands, hundred-thousands) to see how much you're short;
• subtract the second from the first.

Try 328105. The 'one more' digits are 5, 1 and 2: 2 + 1 + 5 = 8. The 'one less' digits are 0, 8 and 3: 3 + 0 + 8 = 11. The difference is 8 − 11 = −3, so the number is 3 short of a multiple of 11 (or, equally, 8 more than one).

A quicker way to write the same thing: put alternating '+' and '−' signs in front of the digits, starting with '+' on the units digit, and add: −3 + 2 − 8 + 1 − 0 + 5 = −3. Same answer, one line. If the result is 0 or a multiple of 11 (not just 11), the number is divisible by 11.

What about numbers with no test of their own? For 6, check its factors 2 and 3; that works, because 2 and 3 share no factors.

But be careful. For 24 it's tempting to check 4 and 6, and that fails: 12 is divisible by both 4 and 6, but not by 24. Check 3 and 8 instead. The reason is in the prime factorisations: 24 = 2 × 2 × 2 × 3, and 3 × 8 supplies all four primes, whereas 4 = 2 × 2 and 6 = 2 × 3 share a 2 and between them guarantee only 12.