2019 AMC 10A Problems/Problem 11

How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)?

${\textbf{(A) }32} \qquad {\textbf{(B) }36} \qquad {\textbf{(C) }37} \qquad {\textbf{(D) }39} \qquad {\textbf{(E) }41}$

Solution 1

Prime factorizing $201^9$ , we get $3^9\cdot67^9$ . A perfect square must have even powers from its prime factors, so our possible choices for our exponents of a perfect square are $0, 2, 4, 6, 8$ for both $3$ and $67$ . This yields $5\cdot5 = 25$ perfect squares. Perfect cubes must have multiples of 3 for each of their prime factors' exponents, so we have either $0, 3, 6$ , or $9$ for both $3$ and $67$ , which yields $4\cdot4 = 16$ perfect cubes. In total, we have $25+16 = 41$ . However, we have overcounted perfect 6'ths: $3^0\cdot67^0$ , $3^0\cdot67^6$ , $3^6\cdot67^0$ , and $3^6\cdot67^6$ . We must subtract these 4, for our final answer, which is $41-4 = \boxed{\textbf{(C) }37}$ .

Solution by Aadileo

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2019 AMC 10A Problems/Problem 11

Solution 1