Divisibility by 11
Place values are alternately 1 more and 1 less than a multiple of 11.
Place values and 11
Look at the place values 1, 10, 100, 1000 next to multiples of 11. 100 = 99 + 1, and 99 is 11 × 9. 10 = 11 − 1.
What happens along 1, 10, 100, 1000, 10000 … ?
What this lesson covers
The idea
Place values are alternately 1 more and 1 less than a multiple of 11, so placing alternating '+' and '–' signs before the digits, starting with '+' at the units digit, and adding gives the remainder on division by 11.
Place values and 11
Look at the place values 1, 10, 100, 1000 next to multiples of 11. 100 = 99 + 1, and 99 is 11 × 9. 10 = 11 − 1.
What happens along 1, 10, 100, 1000, 10000 … ?
- Always 1 more than a multiple of 11
- Alternately 1 more and 1 less
- No pattern
Plus, minus, plus, minus
Every other place value is 1 more (blue, +) or 1 less (orange, −) than a multiple of 11. Add the digits with their signs. Change digits with ▲ ▼ and watch the total.
Alternating signs
Place values are alternately 1 more and 1 less than a multiple of 11, so placing alternating '+' and '–' signs before the digits, starting with '+' at the units digit, and adding gives the remainder on division by 11.
The place values: 1 = 0 + 1 10 = 11 − 1 100 = 99 + 1 1000 = 1001 − 1 and so on.
For 328105: −3 + 2 − 8 + 1 − 0 + 5 = −3
A total of −3 means 3 less than a multiple of 11, which is 8 more than the multiple before it. The remainder is 8. A total of 0 means the number is divisible by 11.
Notes
Place values are alternately 1 more and 1 less than a multiple of 11, so placing alternating '+' and '–' signs before the digits, starting with '+' at the units digit, and adding gives the remainder on division by 11.
Check yourself
Put signs + − + − before the digits of 4075, starting with + at the units digit. What is the total?
Answer: -6
+5 − 7 + 0 − 4 = −6.
So what remainder does 4075 leave when divided by 11?
Answer: 5
−6 means 6 less than a multiple of 11, which is 11 − 6 = 5 more than the multiple before it. So the remainder is 5 (4075 = 11 × 370 + 5).
Which of these numbers is divisible by 11?
Which digit ★ makes 3★52 divisible by 11?
Answer: 6
+2 − 5 + ★ − 3 = ★ − 6. This is 0 when ★ = 6. So 3652 = 11 × 332 is divisible by 11.
- 2900. The digits add up to 11, but we need alternating signs: +0 − 0 + 9 − 2 = 7. Not divisible.
- 1352. +2 − 5 + 3 − 1 = −1. That is 1 less than a multiple of 11, so not divisible.
- 4093. +3 − 9 + 0 − 4 = −10. Not divisible.
- 2728 — correct. Yes! +8 − 2 + 7 − 2 = 11, a multiple of 11. So 2728 = 11 × 248.