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Polynomial factoring

Source: AHSME 1986 problem 23

October 1, 2011
algebrapolynomialAMC

Problem Statement

Let N=695+5⋅694+10⋅693+10⋅692+5⋅69+1.N = 69^{5} + 5\cdot 69^{4} + 10\cdot 69^{3} + 10\cdot 69^{2} + 5\cdot 69 + 1. How many positive integers are factors of NN?
<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 5<spanclass=′latex−bold′>(C)</span> 69<spanclass=′latex−bold′>(D)</span> 125<spanclass=′latex−bold′>(E)</span> 216 <span class='latex-bold'>(A)</span>\ 3\qquad<span class='latex-bold'>(B)</span>\ 5\qquad<span class='latex-bold'>(C)</span>\ 69\qquad<span class='latex-bold'>(D)</span>\ 125\qquad<span class='latex-bold'>(E)</span>\ 216
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