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Probability Power is Real

Source: 2015 AMC 12A #24

February 4, 2015
probabilitytrigonometryfunctioncomplex numbersAMC

Problem Statement

Rational numbers aa and bb are chosen at random among all rational numbers in the interval [0,2)[0,2) that can be written as fractions nd\tfrac nd where nn and dd are integers with 1≤d≤51\leq d\leq 5. What is the probability that (cos⁡(aπ)+isin⁡(bπ))4(\cos(a\pi)+i\sin(b\pi))^4 is a real number?
<spanclass=′latex−bold′>(A)</span>350<spanclass=′latex−bold′>(B)</span>425<spanclass=′latex−bold′>(C)</span>41200<spanclass=′latex−bold′>(D)</span>625<spanclass=′latex−bold′>(E)</span>1350<span class='latex-bold'>(A) </span>\dfrac3{50}\qquad<span class='latex-bold'>(B) </span>\dfrac4{25}\qquad<span class='latex-bold'>(C) </span>\dfrac{41}{200}\qquad<span class='latex-bold'>(D) </span>\dfrac6{25}\qquad<span class='latex-bold'>(E) </span>\dfrac{13}{50}
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