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Sum of Odd and Even Positive Integers

Source: 2012 AMC10A Problem #22

February 8, 2012
modular arithmeticnumber theoryDiophantine equationalgebrasystem of equationsAMC

Problem Statement

The sum of the first mm positive odd integers is 212212 more than the sum of the first nn positive even integers. What is the sum of all possible values of nn?
<spanclass=′latex−bold′>(A)</span> 255<spanclass=′latex−bold′>(B)</span> 256<spanclass=′latex−bold′>(C)</span> 257<spanclass=′latex−bold′>(D)</span> 258<spanclass=′latex−bold′>(E)</span> 259 <span class='latex-bold'>(A)</span>\ 255 \qquad<span class='latex-bold'>(B)</span>\ 256 \qquad<span class='latex-bold'>(C)</span>\ 257 \qquad<span class='latex-bold'>(D)</span>\ 258 \qquad<span class='latex-bold'>(E)</span>\ 259
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