Casting out nines

Summary

Casting out nines is a method to check a computation with integers by the remainders modulo $9$. It uses the fact that a natural number is congruent modulo $9$ to the sum of its decimal digits. To check $a \cdot b = c$, one compares the digit sum of $c$ with the product of the digit sums of $a$ and $b$, all modulo $9$. The same works for sums and differences.

Remarks

If the two results are not congruent modulo $9$, the computation is wrong. If they are congruent, the computation can still be wrong. For example, an exchange of two digits in the result is not detected. The method works because of divisibility by 9 and the compatibility of congruences with addition and multiplication.

Explanations

L1 human

Check $46 \cdot 37 = 1702$. The digit sums are $10 \equiv 1$ and $10 \equiv 1$, so the product should be $1 \cdot 1 = 1$ modulo $9$. The digit sum of $1702$ is $10 \equiv 1$, so the check passes. A wrong result such as $1712$ has the digit sum $11 \equiv 2$, so the check finds the error.