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Binomial raised to a large power

Source: AMC 12 A 2002 #24

August 1, 2007
trigonometrygeometryAMCAIMEcalculusintegrationmodular arithmetic

Problem Statement

Find the number of ordered pairs of real numbers (a,b) (a,b) such that (a \plus{} bi)^{2002} \equal{} a \minus{} bi. <spanclass=latexbold>(A)</span> 1001<spanclass=latexbold>(B)</span> 1002<spanclass=latexbold>(C)</span> 2001<spanclass=latexbold>(D)</span> 2002<spanclass=latexbold>(E)</span> 2004 <span class='latex-bold'>(A)</span>\ 1001\qquad <span class='latex-bold'>(B)</span>\ 1002\qquad <span class='latex-bold'>(C)</span>\ 2001\qquad <span class='latex-bold'>(D)</span>\ 2002\qquad <span class='latex-bold'>(E)</span>\ 2004