Lesson Can there be a perfect square ?
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<H2>Can there be a perfect square ?</H2> <H3>Problem 1</H3>Can there be a perfect square whose digits consist of exactly 4 ones, 4 twos and 4 zeros in any order? <B>Solution</B> <pre> Let N = {{{n^2}}} be such a number. Since the sum of digits of the number " N " 4*1 + 4*2 +4*0 = 4 + 8 + 0 = 12 is divisible by 3, it implies that the number N itself is divisible by 3 (the "divisibility by 3 rule"). In turn, it implies that the number " n " itself is divisible by 3. Then the number {{{n^2}}} is divisible by 3^2 = 9; hence, the number N is divisible by 9. But the sum of the digits of the number N, which was calculated above as 12, is not divisible by 9. It contradicts to the "divisibility by 9 rule". Hence, such a number N with assigned properties <U>DOES NOT EXIST</U>. </pre> The proof is completed. ----------------- On divisibility rules by 3 and by 9 see the lessons - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-3-rule.lesson>Divisibility by 3 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-9-rule.lesson>Divisibility by 9 rule</A> in this site. My other lessons in this site on divisibility numbers are - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-2-rule.lesson>Divisibility by 2 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-3-rule.lesson>Divisibility by 3 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-4-rule.lesson>Divisibility by 4 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-5-rule.lesson>Divisibility by 5 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-6-rule.lesson>Divisibility by 6 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-9-rule.lesson>Divisibility by 9 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-10-rule.lesson>Divisibility by 10 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-11-rule.lesson>Divisibility by 11 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Restore-the-omitted-digit-in-a-number-in-a-way-the-number-is-divisible-by-9.lesson>Restore the omitted digit in a number in a way that the number is divisible by 9</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Restore-the-omitted-digit-in-a-number-in-a-way-the-number-is-divisible-by-11.lesson>Restore the omitted digit in a number in a way that the number is divisible by 11</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-circle-level-problems-on-divisibility-numbers.lesson>Math circle level problems on divisibility numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-circle-level-problem-on-restoring-digit-in-the-product-of-two-16-digit-numbers.lesson>Math circle level problem on restoring digit in the product of two 16-digit numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-circle-level-problem-on-finding-remainders.lesson>Math circle level problem on finding remainders</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/OVERVIEW-of-Divisibility-rules-by-2-3-4-5-6-9-10-11.lesson>OVERVIEW of Divisibility rules by 2, 3, 4, 5, 6, 9, 10 and 11</A>
