Remainders of Squares and Cubes
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When we divide a number by m, we get a remainder between 0 and m−1. The remainder depends on the number, but for squares and cubes only certain remainders are possible. These are called the remainders (or residues) of sq…
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When we divide a number by m, we get a remainder between 0 and m−1. The remainder depends on the number, but for squares and cubes only certain remainders are possible. These are called the remainders (or residues) of squares and cubes modulo m.
For example, if we divide any square by 3, the remainder can only be 0 or 1. If we divide any square by 4, the remainder can only be 0 or 1 as well. This idea is useful because sometimes we can prove something is impossible just by checking all possible remainders.