## Perfect Squares ending in 576 [closed]

I want to find out perfect squares ending in 576, after the number 576.

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..

Hence I can write my required perfect square ending in 576 as -

(1+3+5+7+ ....49) + (51+53+55+57+....n terms)

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

(51+53+55+57+....n terms) = 1000k

Using formula for Arithmetic Progression starting with 51 and a common difference of 2,

n/2[2(51) + (n-1)2] = 1000k

n(n+50) = 1000k

Put n = 100, 100*150 = 1000k, hence k = 15.

Put k = 15 in the perfect square term 1000k+576 we get the number 15576.

But 15576 is NOT a perfect square.

What is flawed in my derivation? Kindly help.

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 This question is not suitable for this site. Read the FAQ and perhaps try math.stackexchange.com – Kevin Walker Jan 31 2012 at 15:19 The sum of an odd number of odd numbers is an odd number – Ramiro de la Vega Jan 31 2012 at 15:33 Count the number of terms correctly. Gerhard "Ask Me About System Design" Paseman, 2012.01.31 – Gerhard Paseman Jan 31 2012 at 15:35