CUADRADOS Y CUBOS PERFECTOS - CARACTERÍSTICAS DE EXCLUSIÓN - EJEMPLOS DE POTENCIACIÓN ARITMÉTICA

Exclusionary Characteristics of Perfect Square Numbers, Exclusionary Characteristics of Perfect Cube Numbers I) The square of a number ends in the square of its units digit. II) Any number ending in 0; 1; 4; 5; 6; or 9 can be a perfect square. It follows that if a number ends in 2; 3; 7; or 8, it is not a perfect square. For example: Of the following numbers: (I) 253762; (II) 514384; (III) 519637 ; (IV) 5180048; (V) 6143723; (VI) 516325 are not perfect squares (I); (III); (IV); and (V) The following can be perfect squares: (II) and (VI). III) If a number ends in 5, it can be a perfect square as long as its tens digit is 2 and its hundreds digit is the product of two consecutive numbers. If: , then: that is: c = 0; 2 or 6 If a positive integer ends in 5, but its tens digit is not 2 or its third-order digit is not: 0; 2 or 6, then that number is not a perfect square. This follows from the analysis of the square of any number ending in 5. Therefore, the square of a number ending in 5 always ends in 25. IV) Any number ending in an even number of zeros can be a perfect square, as long as the accompanying digits form a perfect square. If: it is a perfect square. If a number ends in an odd number of zeros, it is not a perfect square. This is known because the square of a number ending in "n" zeros will end in (2n) zeros. Thus: 902 = 8100 12002 = 1440000 But 36000 is not a perfect square because it ends in three zeros. V) Every perfect square is of the form VI) If a positive integer is divisible by a prime number "p" but not divisible by p2 , it is not a perfect square. This is because when factored into prime factors, the number would contain the prime factor "p" raised to the power of one (odd). Example: SOME INTERESTING FACTS ABOUT THE FIGURES OF PERFECT SQUARES I) If the integers 1, 2, 3, 4, 5, 6, 7, 8, 9 ... are squared 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, ... the following law is observed: The unit figures of these squares form a symmetric period. 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, .... with equal digits relative to 5 or 0. II) The last two digits of consecutive squares form a period of 51 numbers. 00, 01, 04, 09, 16, ... , 76, 25, 76, ..., 16, 09, 04, 01, 00 symmetrical relative to 25 or 00. This observation extends indefinitely. The last three digits of consecutive perfect squares form a period of 501 numbers. The last four form a period of 2501 numbers, etc. III) There are some squares that are written with all different digits. Example: 132 = 169 10272 = 1054729 362 = 1296 69012 = 47623801 2862 = 81796 101242 = 102495376 3222 = 103684 320432 = 1026753849 IV) The pairs of perfect squares: 144 and 441; 169 and 961; 14884 and 48841 and their respective roots: 12 and 21; 13 and 31; 122 and 221, are formed by the same numbers, but written in reverse order.