# Multiplying Complementary Pairs

Quick! What’s 23 x 27?

621

There’s a trick to doing this quickly. Can you see a pattern in these multiplications?

42 x 48 = 2016
43 x 47 = 2021
44 x 46 = 2024
54 x 56 = 3024
64 x 66 = 4224
61 x 69 = 4209
111 x 119 = 13209

In each pair above, the numbers being multiplied are complementary: they are the same number except for the rightmost digit, and the rightmost digits add to 10.

The trick to multiplying complementary pairs is to take the rightmost digits and multiply them; the result forms the two rightmost digits of the answer. (So in the last example 1 x 9 = 09.) Then take the first number without its rightmost digit, and multiply it by the next higher whole number; the result forms the initial digits of the answer. (So in the last example: 11 x 12 = 132. Voila! The answer is 13209.)

The Math Behind the Fact:
This trick works because you are multiplying pairs of numbers of the form 10*N+A and 10*(N+1)-A, where N is a whole number and A is a digit between 1 and 9. A little shows their product is:

100*N*(N+1) + A*(10-A).

The first term in the sum is a multiple of 100 and it does not interact with the last two digits of sum, which is never more than two digits long.

This trick is related to some of the  lightning  Fun Facts.

Su, Francis E., et al. “Multiplying Complementary Pairs.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
A. Benjamin and M. Shermer, Secrets of Mental Math, Three Rivers Press, 2006.

Fun Fact suggested by:
Francis Su

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