Casting out nines

Text

Casting out nines checks a multiplication by comparing the remainders on division by $9$ (obtained by repeatedly summing the digits). Apply casting out nines to the product $2137\times 46 = 98302$: compute the remainder modulo $9$ of the two factors and of the result, and check that they match.

Solution

Sum the digits (the remainder modulo $9$ is unchanged):

• $2137 \to 2+1+3+7 = 13 \to 1+3 = 4$
• $46 \to 4+6 = 10 \to 1$
• product of the remainders: $4\cdot 1 = 4$
• $98302 \to 9+8+3+0+2 = 22 \to 2+2 = 4$

The two values agree ($4=4$): the check passes, with common remainder $\boxed{4}$. (Note: the check does not guarantee correctness, but a failure certainly signals an error.)