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==Odd and even square numbers== <!-- This section is linked from [[Irrational number]] --> [[File:visual_proof_centered_octagonal_numbers_are_odd_squares.svg|thumb|upright|[[Proof without words]] that all centered octagonal numbers are odd squares]] Squares of even numbers are even, and are divisible by 4, since {{Math|1=(2''n'')<sup>2</sup> = 4''n''<sup>2</sup>}}. Squares of odd numbers are odd, and are congruent to 1 [[modular arithmetic|modulo]] 8, since {{Math|1=(2''n'' + 1)<sup>2</sup> = 4''n''(''n'' + 1) + 1}}, and {{Math|''n''(''n'' + 1)}} is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8. Every odd perfect square is a [[centered octagonal number]]. The difference between any two odd perfect squares is a multiple of 8. The difference between 1 and any higher odd perfect square always is eight times a triangular number, while the difference between 9 and any higher odd perfect square is eight times a triangular number minus eight. Since all triangular numbers have an odd factor, but no two values of {{math|2<sup>''n''</sup>}} differ by an amount containing an odd factor, the only perfect square of the form {{math|2<sup>''n''</sup> β 1}} is 1, and the only perfect square of the form {{math|2<sup>''n''</sup> + 1}} is 9.
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