D. Factors
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The fundamental theorem of arithmetic states that every integer greater than 1 can be uniquely rep\(re- se\)nted as a product of one or more primes. While unique, several arrangements of the prime factors may be possible. For example: \(10 = 2 \cdot 5 20 = 2 \cdot 2 \cdot 5 = 5 \cdot 2 = 2 \cdot 5 \cdot 2 = 5 \cdot 2 \cdot 2\) Let \(f\)(\(k\)) be the number of different arrangements of the prime factors of \(k\). So \(f\)(10) = 2 and \(f\)(20) = 3. Given a positive number \(n\), there always exists at least one number \(k\) such that \(f\)(\(k\)) = \(n\). We want to know the smallest such \(k\).
The input consists of at most 1 000 test cases, each on a separate line. Each test case is a positive integer \(n < 2^{63}\).
For each test case, display its number \(n\) and the smallest number \(k > 1\) such that \(f\)(\(k\)) = \(n\). The numbers in the input are chosen such that \(k < 2^{63}\).
1
2
3
105
1 2
2 6
3 12
105 720
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D. Factors
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D. Factors