Perfect Squares ending in 576 - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-21T07:22:34Z http://mathoverflow.net/feeds/question/87140 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87140/perfect-squares-ending-in-576 Perfect Squares ending in 576 unknown (google) 2012-01-31T15:15:37Z 2012-01-31T15:15:37Z <p>I want to find out perfect squares ending in 576, after the number 576.</p> <p>Here is my derivation to arrive at such a number. Let the perfect square ending in 576 be 1000k+576. Every perfect square can be expressed as a the sum of a certain number of consecutive odd numbers. For eg: 2^2 = 1+3, 3^2 = 1+3+5, 4^2 = 1+3+5+7, and so on..</p> <p>Hence I can write my required perfect square ending in 576 as -</p> <p>(1+3+5+7+ ....49) + (51+53+55+57+....n terms) </p> <p>Therefore, (1+3+5+7+ ....49) + (51+53+55+57+....n terms) = 1000k +576. Since, (1+3+5+7+ ....49) = 576, the equation reduces to</p> <p>(51+53+55+57+....n terms) = 1000k</p> <p>Using formula for Arithmetic Progression starting with 51 and a common difference of 2, </p> <p>n/2[2(51) + (n-1)2] = 1000k</p> <p>n(n+50) = 1000k</p> <p>Put n = 100, 100*150 = 1000k, hence k = 15.</p> <p>Put k = 15 in the perfect square term 1000k+576 we get the number 15576.</p> <p>But 15576 is NOT a perfect square.</p> <p>What is flawed in my derivation? Kindly help.</p>