It freaks me out!

Number Theory Level pending

If S S is the sum of all positive integers satisfying x x in the equation below:

x 2 + ( x + 1 ) 2 + ( x + 2 ) 2 + + ( x + 500 ) 2 = ( x + 501 ) 2 + ( x + 502 ) 2 + + ( x + 1000 ) 2 x^{2} + (x+1)^{2} + (x+2)^{2}+ \cdots +(x+500)^{2} = (x+501)^{2} + (x+502)^{2} +\cdots + (x+1000)^{2}

Find S m o d 9 S \bmod 9 .


The answer is 1.

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1 solution

Otto Bretscher
Jan 7, 2016

As we show here , the only positive solution is x = k ( 2 k + 1 ) = 500 × 1001 1 ( m o d 9 ) x=k(2k+1)=500\times 1001\equiv \boxed{1} \pmod9

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