## Parity ArgumentsIf two integers are either both even or both odd, they are said to have the same parity; otherwise they have different parity. Determining the parity of two quantities is often a simple and useful way to prove that the quantities can never be equal. That result, in turn, can be used to demonstrate that a particular situation is impossible. Parity is just a special case of divisibility. Although we do not have
special words for “divisible by 5” or “leaves a remainder
when divided by 7,” issues of divisibility arise frequently. For
example, a fourth grader was investigating which
The student conjectured that at least one of the dimensions of the
rectangle had to be divisible by 5 for a tiling to be possible. She
explained that the area of the pentomino had to divide evenly into the
area of the rectangle. Since 5 is prime, the only way the rectangle’s
area, For practice with proofs that involve parity arguments, see Parity Problems and Parity Problems 2. For practice with, and further information about, divisibility, see Divisibility and Modular Arithmetic. |

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