Multiplying by 11
Three moves: write the first digit, add each pair of neighbours, write the last digit. Carries are resolved in place, so you never multiply anything by 11.
1. The three-step rule
- Last digit: copy the last digit of the multiplicand.
- Sums: moving left, add each pair of neighbours (this digit + the one on its right). If a sum reaches 10 or more, write the units digit and carry 1 into the next sum.
- First digit: copy the leading digit, adding any carry from the previous step.
Why it works
Multiplying by 11 is multiplying by (10 + 1): each digit lands in its own place and once more one place to the left. The neighbour sum is exactly the overlap of the two shifted copies — which is why carries only ever move one position left.
2. Worked example
428 × 11
- Last digit: 8
- 2 + 8 = 10 → write 0, carry 1
- 4 + 2 = 6, plus the carry = 7
- Leading digit 4, no carry left → 4
Reading left to right: 4708.
With a carry chain: 89 × 11 → last 9; 8+9 = 17 → write 7 carry 1; 8 + 1 = 9 → 979.
3. Live demo
Type any integer and watch the full derivation
The demo is computed in your browser; nothing is sent anywhere.
4. Practice
- 52 × 11
- 89 × 11
- 316 × 11
- 1234 × 11
- 405 × 11 — mind the carry through the zero
Answers
1) 572. 2) 979. 3) 3476. 4) 13574. 5) 4455 (0+5 = 5, 4+0 = 4, leading 4 → 4 4 5 5).
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