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 231 243. 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 6. Adding the digits of 243 together, we get 9. Multiply 6 9 to get 54, 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. Let’s start with the following definition.
Given a positive integer , define its digit sum to be the sum of the digits in its base 10 representation, with the stipulation that if this sum is 10 or larger then the process of adding digits is applied repeatedly until a single digit is obtained.
Then the trick my friend showed me claims that the following is true:
Theorem 1: .
The most important piece of the proof of Theorem 1 is that is actually calculating the remainder when is divided by 9. Using standard notation for remainders, let’s call this
Lemma 1: .
Proof of Lemma 1: Let the base 10 representation of be given by . Then . (The last step is valid because , so we can replace with in the sum.) Thus calculating gives the same result as calculating the sum of its digits mod 9. If (i.e., the sum of the digits of ), finding the sum, mod 9, of the digits of gives the same result as . Continuing this process shows that can be determined by calculating the sum of the digits of , calculating the sum of the digits of that result, and so forth. This continues until a single digit is obtained, which must be the value of . Since in the case where has a single digit in base 10, and follows exactly the process described here when consists of two or more digits in base 10, it must be the case that .
Back to the proof of Theorem 1: With Lemma 1 in mind, we can rewrite the claim in Theorem 1 like so:
This is a standard property of modular arithmetic, and so we could stop here, but let’s prove it for the sake of completeness. Let , where is the quotient when is divided by 9, and is the remainder. Similarly, let .
Then , as the other terms are divisible by 9.
Similarly, proving the theorem.
This check works for addition and subtraction as well. For more on that, see this site on the arithmetic of digit sums.
The check can be simplified further by removing, at any point in the process, a 9 that appear in a digit sum. This should make sense, given that finding a digit sum is just working mod 9. This simplification is called “casting out nines.”
Of course, the check isn’t infallible. It’s possible to make a mistake in the multiplication that yields the same result mod 9. However, the chances of that (assuming a uniform distribution on the mod 9 result when a mistake is made) is only 1/9.