Question

If r and s are positive integers, can the fraction $\frac{r}{s}$ be expressed as a decimal with only a finite number of nonzero digits?
(1) $s$ is a factor of $100$.
(2) $r$ is a factor of $100$.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is A Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

Determine if where r and s are positive integers, can be expressed as a decimal with a finite number of nonzero decimal digits.

1. It is given that s is a factor of 100 and so s = 1, 2, 4, 5, 10, 20, 25, 50, or 100. This means that must be one of the quotients or . Then, must be one of the products r(1), r(0.5), r(0.25), r(0.20), r(0.1), r(0.05), r(0.04), r(0.02), or r(0.01). In each case, is the product of an integer and a decimal with a finite number of nonzero digits, and hence, can be expressed as a decimal with a finite number of nonzero digits. In fact, it suffices to note this is true for , since each of the other possibilities is a positive integer times ; Thus, sufficient.
2. It is given that r is a factor of 100. If r = 4 and s = 5, then , which is a decimal with a finite number of nonzero digits. On the other hand, if r = 4 and s = 7, then , which is not a decimal with a finite number of nonzero digits; NOT sufficient.

• The correct answer is A; statement 1 alone is sufficient.