Some multiplications are particularly easy to do mentally. Since 11 = 10 + 1, multiplying any integer N by 11 is very quick. For example, append a 0 to N, then add N to the resulting number. In most situations, however, another technique proves more efficient. If N has only two digits, add them together and insert the result between them. Thus, 72 × 11 = 792. Remember to carry, adding the carried digit to the hundreds digit. For example, 79 × 11 = 869.
This technique extends immediately to integers with more than two digits: working from right to left, add adjacent pairs of digits and insert each result in turn, remembering to carry. Take 86,509 × 11. The units digit is 9. The tens digit is 0 + 9 = 9, so we insert 9 to obtain 99. The hundreds digit is 5 + 0 = 5; inserting 5 gives 599. The thousands digit is 6 + 5 = 11, so we insert 1 and carry 1, giving 1,599. The ten-thousands digit is 8 + 6 + 1 = 15, so we write 5 and carry 1, giving 51,599. Finally, the hundred-thousands digit is 8 + 1 = 9. The result is 951,599. Easy!
Iterative methods
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The little mental workout involved in multiplying an integer N by 11 in your head can be unpacked into a test for divisibility by 11. You may have learned this technique at school. It is called the domino divisibility test for 11:
An integer is divisible by 11 if and only if the positive difference between its number of tens and its units digit is a multiple of 11.
Thus, 781 is divisible by 11 because 78 – 1 = 77, which is clearly a multiple of 11. By contrast, 459 is not divisible by 11 because 45 – 9 = 36, which is not a multiple of 11.
