Using this shortcut, find out whether the following numbers are divisible by 11. Further, find the remainder if the number is not divisible by 11.
158
158 is not divisible by 11. The remainder is 4. Calculation: -1 + 5 - 8 = -4, which means 158 is 4 less than a multiple of 11.
841
841 is not divisible by 11. The remainder is 5. Calculation: -8 + 4 - 1 = -5, which means 841 is 5 less than a multiple of 11.
481
481 is not divisible by 11. The remainder is 8. Calculation: -4 + 8 - 1 = 3, which means 481 is 8 more than a multiple of 11, so remainder is 8.
5529
5529 is not divisible by 11. The remainder is 7. Calculation: -5 + 5 - 2 + 9 = 7, which means 5529 is 7 more than a multiple of 11.
90904
90904 is divisible by 11. Calculation: 9 - 0 + 9 - 0 + 4 = 22, which is divisible by 11.
857076
857076 is divisible by 11. Calculation: -8 + 5 - 7 + 0 - 7 + 6 = -11, which is divisible by 11.
Explanation
(i) Using the alternating signs method: -1 + 5 - 8 = -4. Since -4 is not 0 or divisible by 11, 158 is not divisible by 11. The result -4 means 158 is 4 less than a multiple of 11, so the remainder is 4. (ii) Using the alternating signs method: -8 + 4 - 1 = -5. Since -5 is not 0 or divisible by 11, 841 is not divisible by 11. The remainder is 5. (iii) Using the alternating signs method: -4 + 8 - 1 = 3. Since 3 is not 0 or divisible by 11, 481 is not divisible by 11. The result 3 means 481 is 8 more than a multiple of 11, so the remainder is 8. (iv) Using the alternating signs method: -5 + 5 - 2 + 9 = 7. Since 7 is not 0 or divisible by 11, 5529 is not divisible by 11. The remainder is 7. (v) Using the alternating signs method starting from units: -9 + 0 - 9 + 0 - 4 = -22. Since -22 is divisible by 11, 90904 is divisible by 11. (vi) Using the alternating signs method: -8 + 5 - 7 + 0 - 7 + 6 = -11. Since -11 is divisible by 11, 857076 is divisible by 11.