Quick! What’s 23 x 27? <\/p>\n\n\n\n
621<\/p>\n\n\n\n
There’s a trick to doing this quickly. Can you see a pattern in these multiplications? <\/p>\n\n\n\n
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<\/p>\n\n\n\n
In each pair above, the numbers being multiplied are complementary<\/em>: they are the same number except for the rightmost digit, and the rightmost digits add to 10.<\/p>\n\n\n\n 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.)<\/p>\n\n\n\n The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong> 100*N*(N+1) + A*(10-A).<\/p>\n\n\n\n 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.<\/p>\n\n\n\n This trick is related to some of the other\u00a0lightning arithmetic\u00a0Fun Facts.<\/p>\n\n\n\n How to Cite this Page:<\/strong>
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: <\/p>\n\n\n\n
Su, Francis E., et al. “Multiplying Complementary Pairs.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n