Smallest positive Integer Problem | AIME I, 1990 | Question 5

Let n be the smallest positive integer that is a multiple of 75 and has exactly 75 positive integral divisors, including 1 and itself. Find \(\frac{n}{75}\).

75=\(3 \times 5^{2}\)=(2+1)(4+1)(4=1)

or, \(n=p_1^{a_1-1}p_2^{a_2-1}.....\) such that \(a_1a_2....=75\)

or, 75|n with two prime factors 3 and 5

Minimizing n third factor =2

and factor 5 raised to least power

or, \(n=(2)^{4}(3)^{4}(5)^{2}\)

and \(\frac{n}{75}=(2)^{4}(3)^{4}(5)^{2}\)=(16)(27)=432.

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