M03-31

Official Solution:


THEORY:

A reduced fraction \(\frac{a}{b}\) (meaning that the fraction is already in its simplest form, so reduced to its lowest term) can be expressed as a terminating decimal if and only if the denominator \(b\) is of the form \(2^n5^m\), where \(m\) and \(n\) are non-negative integers. For example: \(\frac{7}{250}\) is a terminating decimal \(0.028\), as the denominator \(250\) equals \(2*5^3\). The fraction \(\frac{3}{30}\) is also a terminating decimal, as \(\frac{3}{30}=\frac{1}{10}\) and the denominator \(10=2*5\).

Note that if the denominator already consists of only 2s and/or 5s, then it doesn't matter whether the fraction is reduced or not.

For example, \(\frac{x}{2^n5^m}\), (where \(x\), \(n\), and \(m\) are integers) will always be a terminating decimal.

(We need to reduce the fraction in case the denominator has a prime other than 2 or 5, to see whether it can be reduced. For example, the fraction \(\frac{6}{15}\) has 3 as a prime in the denominator, and we need to know if it can be reduced.)

BACK TO THE ORIGINAL QUESTION:

If \(r\) and \(s\) are positive integers, is \(\frac{r}{s^2}\) a terminating decimal?

(1) \(s = 225\).

Hence \(\frac{r}{s^2}=\frac{r}{225^2}=\frac{r}{3^4*5^4}\). We don't know whether \(3^4\) can be reduced, so we cannot determine whether this fraction will be a terminating decimal. Not sufficient.

(2) \(r = 81\). This information is not sufficient on its own.

(1)+(2) \(\frac{r}{s^2}=\frac{3^4}{3^4*5^4}=\frac{1}{5^4}\). Since the denominator has only 5 as a prime, this fraction is a terminating decimal. Both statements together are sufficient.


Answer: C