M12-19

Official Solution:

How many positive factors of 72 are divisible by 2?

A. 4
B. 5
C. 6
D. 8
E. 9


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:

Using the method above, since \(72=2^3*3^2\), the total number of factors of 72 is \((3+1)(2+1) = 12\). Of these, only three are odd: 1, 3, and 9. This is because the only factors of 72 that are odd arise from powers of 3. The number of odd factors for \(3^2\) is \((2 + 1) = 3\), specifically - 1, 3, and 9. Thus, the other \(12-3=9\) factors are even.

OR: Since \(72=2^3*3^2\), even factors MUST have 2 raised to either the power of 1, 2, or 3, providing 3 options, and 3 raised to the power of 0, 1, or 2, again giving 3 options. This results in a total of \(3*3=9\) even factors.


Answer: E