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Square number
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==Special cases== * If the number is of the form {{math|''m''5}} where {{math|''m''}} represents the preceding digits, its square is {{math|''n''25}} where {{math|1=''n'' = ''m''(''m'' + 1)}} and represents digits before 25. For example, the square of 65 can be calculated by {{math|1=''n'' = 6 Γ (6 + 1) = 42}} which makes the square equal to 4225. * If the number is of the form {{math|''m''0}} where {{math|''m''}} represents the preceding digits, its square is {{math|''n''00}} where {{math|1=''n'' = ''m''<sup>2</sup>}}. For example, the square of 70 is 4900. * If the number has two digits and is of the form {{math|5''m''}} where {{math|''m''}} represents the units digit, its square is {{math|''aabb''}} where {{math|1=''aa'' = 25 + ''m''}} and {{math|1=''bb'' = ''m''<sup>2</sup>}}. For example, to calculate the square of 57, {{Math|1=''m'' = 7}} and {{Math|1=25 + 7 = 32}} and {{Math|1=7<sup>2</sup> = 49}}, so {{Math|1=57<sup>2</sup> = 3249}}. * If the number ends in 5, its square will end in 5; similarly for ending in 25, 625, 0625, 90625, ... 8212890625, etc. If the number ends in 6, its square will end in 6, similarly for ending in 76, 376, 9376, 09376, ... 1787109376. For example, the square of 55376 is 3066501376, both ending in ''376''. (The numbers 5, 6, 25, 76, etc. are called [[automorphic number]]s. They are sequence [[oeis:A003226|A003226]] in the [[On-Line Encyclopedia of Integer Sequences|OEIS]].<ref>{{Cite OEIS|1=A003226|2=Automorphic numbers: n^2 ends with n.}}</ref>) * In base 10, the last two digits of square numbers follow a repeating pattern mirrored symmetrical around multiples of 25. In the example of 24 and 26, both 1 off from 25, {{Math|1=24<sup>2</sup> = 576}} and {{Math|1=26<sup>2</sup> = 676}}, both ending in 76. In general, <math display="inline">(25n+x)^2-(25n-x)^2=100nx</math>. An analogous pattern applies for the last 3 digits around multiples of 250, and so on. As a consequence, of the 100 possible last 2 digits, only 22 of them occur among square numbers (since 00 and 25 are repeated).
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