M01-35

Official Solution:

For function \(\#\), \(\#x\) represents the number of distinct positive divisors of the positive integer \(x\). What is the value of \(\#(\#90)\)?

A. 3
B. 4
C. 5
D. 6
E. 7


Finding the Number of Factors of an Integer

First, make the prime factorization of an integer \(n = a^p * b^q * c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\), and \(p\), \(q\), and \(r\) are their respective powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and \(n\) itself.

Example: Finding the number of all factors of 450: \(450 = 2^1 * 3^2 * 5^2\)

The total number of factors of 450, including 1 and 450 itself, is \((1+1)(2+1)(2+1) = 2*3*3 = 18\) factors.

Back to the original question:

The question defines \(\#x\) as the number of distinct positive divisors of \(x\). For example, \(\#6=4\), as 6 has 4 distinct positive divisors: 1, 2, 3, and 6.

Question: \(\#(\#90)=\)?

Since \(90 = 2 * 3^2 * 5\), the number of factors of 90 is: \((1+1)(2+1)(1+1) = 12\). So \(\#90 = 12\). Next, we need to find \(\#(\#90) = \#12\). Now, as \(12 = 2^2 * 3\), the number of factors of 12 is: \((2+1)(1+1) = 6\).


Answer: D