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Counting Divisors

Source: 2016 AMC10A #22

February 3, 2016
2016 AMC 10ADivisors2016 AMC 12AAMC 10AAMC 12AAMC 12number theory

Problem Statement

For some positive integer nn, the number 110n3110n^3 has 110110 positive integer divisors, including 11 and the number 110n3110n^3. How many positive integer divisors does the number 81n481n^4 have?
<spanclass=′latex−bold′>(A)</span>110<spanclass=′latex−bold′>(B)</span>191<spanclass=′latex−bold′>(C)</span>261<spanclass=′latex−bold′>(D)</span>325<spanclass=′latex−bold′>(E)</span>425<span class='latex-bold'>(A) </span>110 \qquad <span class='latex-bold'>(B) </span> 191 \qquad <span class='latex-bold'>(C) </span> 261 \qquad <span class='latex-bold'>(D) </span> 325 \qquad <span class='latex-bold'>(E) </span> 425
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