M02-24

Official Solution:

How many positive integers are factors of 90?

A. 6
B. 8
C. 9
D. 10
E. 12


Determining the Number of Positive Factors of a Positive Integer \(n\):

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

The number of positive factors of \(n\) can be determined by the formula \((a+1)(b+1)(c+1)\). NOTE: This includes 1 and \(n\) itself.

For example, to find the number of positive factors of 450, we first perform its prime factorization: \(450 = 2^1 * 3^2 * 5^2\). The total number of positive factors of 450, including 1 and 450, is \((1+1)(2+1)(2+1) = 18\) positive factors.

According to the above, 90, which is equal to \(2*3^2*5\), has \((1+1)(2+1)(1+1) = 12\) positive factors, namely 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.


Answer: E