A Narrow Margin - Checking Multiplication via Digit Sums

Posted on December 27, 2011

Last week a friend who is a fourth grade teacher came to me with a math problem. The father of one of his students had showed him a trick for checking the result of a three-digit multiplication problem. The father had learned the trick as a student himself, but he didn’t know why it worked. My friend showed me the trick and asked if I had seen it before. This post describes this check and explains why it works.

Suppose you want to multiply 231 times 243. Working it out by hand, you get 56133. Add the digits in the answer (5+6+1+3+3) to get 18. Add the digits again to get 9. Stop now that you have a single digit.

Alternatively, do this digit adding beforehand. Adding the digits of 231 together, we get 7. Adding the digits of 243 together, we get 9. Multiply 7 times 9 to get 63, then add these digits to get 9. We got 9, just as we did before. And that’s the check: You have to get the same thing with either process.

This almost seems like magic, but there’s some interesting basic number theory going on here.

http://mikespivey.wordpress.com/2011/12/27/digit-sums/