12 Days of Christmas GMAT Competition - Day 10: Any decimal that has

GMAT Club Official Explanation:

Any decimal that has only a finite number of nonzero digits is a terminating decimal. For example, 24, 0.82, and 5.096 are three terminating decimals. If s is a positive integer and the ratio 1/s is expressed as a decimal, is 1/s a terminating decimal?

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.)

(1) s! ends with exactly one is 0

This implies that s! has only one factor of 5, hence s can be 5, 6, 7, 8, or 9. If s is 5 or 8, 1/s will be a terminating decimal. However, if s is 6, 7, or 9, 1/s will NOT be a terminating decimal. Not sufficient.

(2) The sum of any two positive factors of s is even

This implies that all factors of s are odd, so s itself is odd. If s is a power of 5: 5, 25, ... and so on, 1/s will be a terminating decimal. However, if s is any other odd number (greater than 1), 1/s will NOT be a terminating decimal. Not sufficient.

(1)+(2) Since from (2) s is odd, then from (1) it can be 5, 7, or 9. If s is 5, 1/s will be a terminating decimal. However, if s is 7, or 9, 1/s will NOT be a terminating decimal. Not sufficient.

Answer: E.