The remainders when three positive integers are divided by 5 are 1, 2, and 3.  Find the remainder when their product is divided by 5.
I need to find the product of the three integers modulo 5, which means I only care about their remainders modulo 5. Multiplying the remainders together, I get 1 * 2 * 3 = 6. But 6 is not a valid remainder modulo 5, since it is greater than 5. So I need to reduce 6 modulo 5, which means finding the smallest non-negative integer that differs from 6 by a multiple of 5. I can subtract 5 from 6 to get 1, which is the smallest such integer. So the remainder when the product is divided by 5 is 1.

