Problem Solver's Toolbox
This will be a page for the Wisconsin Math Talent Search where we will collect simple problem solving strategies.
When we have divide two integers, they don't always divide evenly, and there is a quotient and a remainder. For example when we divide 10 by 3 we get a remainder of 1. It turns out that these remainders behave very well under addition, subtraction, and multiplication. We say two numbers are the same "modulo m" if they have the same remainder when divided by m. If a and a' are the same modulo m, and b and b' are the same modulo m, then a+b and a'+b' are the same modulo m, and similarly for subtraction and multiplication. This often makes calculation much simpler. For example, see 2016-17 Set #2 problem 3.
See Art of Problem Solving's introduction to modular arithmetic for more information.