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 algebra 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. -*
