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P26 [Number Theory] - Turkish NMO 1st Round - 2013

Source:

April 18, 2013
modular arithmeticnumber theorygreatest common divisor

Problem Statement

What is the maximum number of primes that divide both the numbers n3+2n^3+2 and (n+1)3+2(n+1)^3+2 where nn is a positive integer?
<spanclass=latexbold>(A)</span> 3<spanclass=latexbold>(B)</span> 2<spanclass=latexbold>(C)</span> 1<spanclass=latexbold>(D)</span> 0<spanclass=latexbold>(E)</span> None of above <span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 1 \qquad<span class='latex-bold'>(D)</span>\ 0 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}