## If $$r$$ and $$s$$ are positive integers, can the fraction $$\dfrac{r}{s}$$ be expressed as a decimal with only a finite

##### This topic has expert replies
Legendary Member
Posts: 1441
Joined: 01 Mar 2018
Followed by:2 members

### If $$r$$ and $$s$$ are positive integers, can the fraction $$\dfrac{r}{s}$$ be expressed as a decimal with only a finite

by Gmat_mission » Sun Sep 19, 2021 12:41 pm

00:00

A

B

C

D

E

## Global Stats

If $$r$$ and $$s$$ are positive integers, can the fraction $$\dfrac{r}{s}$$ 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 $$100.$$

Source: Official Guide

### GMAT/MBA Expert

GMAT Instructor
Posts: 15795
Joined: 08 Dec 2008
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1267 members
GMAT Score:770

### Re: If $$r$$ and $$s$$ are positive integers, can the fraction $$\dfrac{r}{s}$$ be expressed as a decimal with only a fi

by [email protected] » Sun Sep 19, 2021 2:24 pm

00:00

A

B

C

D

E

## Global Stats

Gmat_mission wrote:
Sun Sep 19, 2021 12:41 pm
If $$r$$ and $$s$$ are positive integers, can the fraction $$\dfrac{r}{s}$$ 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 $$100.$$

Source: Official Guide
Target question: Is r/s a terminating decimal?

Statement 1: s is a factor of 100
There's a nice rule that says something like,
If the prime factorization of the denominator contains only 2's and/or 5's, then the decimal version of the fraction will be a terminating decimal.
Since 100 = (2)(2)(5)(5), any factor of 100 will contain only 2's and/or 5's (or the denominator can be 1, in which case the decimal will definitely terminate).
Since the denominator of r/s must contain only 2's and/or 5's, r/s must be a terminating decimal
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: r is a factor of 100
There are several pairs of values that meet this condition. Here are two:
Case a: r = 1 and s = 4, in which case r/s = 1/4 = 0.25, which is a terminating decimal
Case b: r = 1 and s = 3, in which case r/s = 1/3 = 0.333.., which is not a terminating decimal
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT