Maths Olympiad Prep

Track / Stage 3 / 36 of 260 #36 of 1964

Problem 36

AMC 10/12, early questions
Number theory Difficulty 3.1 Multiple choice

The number of distinct positive integral divisors of (30)4(30)^4 excluding 11 and (30)4(30)^4 is

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Official solution

Theprimefactorizationof$30$is$2⋅3⋅5$,sotheprimefactorizationof$304$is$24⋅34⋅54$.Therefore,thenumberofpositivedivisorsof$304$is$(4+1)(4+1)(4+1)=125$.However,wehavetosubtract$2$toaccountfor$1$and$304$,soourfinalansweris$125−2=123,C The prime factorization of \$30\$ is \$2\cdot3\cdot5\$, so the prime factorization of \$30^4\$ is \$2^4\cdot3^4\cdot5^4\$. Therefore, the number of positive divisors of \$30^4\$ is \$(4+1)(4+1)(4+1)=125\$. However, we have to subtract \$2\$ to account for \$1\$ and \$30^4\$, so our final answer is \$125-2=123, \boxed{\text{C}}

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.