If r and s are positive integers, can the fraction r/s be expressed as

If r and s are positive integers, can the fraction r/s be expressed as a decimal with only a finite number of nonzero digits?

We can re frame the question as is r/s a terminating decimal or not?

if the only prime factors of s are either 2 or 5 or both 2 and 5, then we can conclude that r/s will terminating.

For example: 1/5, 11/32, 7/25, 17/50 are all terminating decimals. When we prime factorize the denominators we see that the only prime factors are either 2 or 5 or both.

If s has prime factors other than 2 and 5 , then we need to check if its still there as the prime factor of denominator of the reduced/simplified form of r/s. If yes, then its a non terminating decimal.

For example: 3/14, 5/22, 13/70 are non terminating decimal.

3/14 = 3/2*7 is non terminating as 7 is a prime factor of the denominator.
5/22= 5/ 2*11 is a non terminating decimal as 11 is a prime factor of the denominator.

lets try an example 14/35 .
14/35 =14/(5*7)
Since denominator have a prime factor of 7, we need to cross check it's there in the simplified form as well.
Simplified fraction is 2/5 =.4 which is terminating decimal as 7 get cancelled in the numerator and denominator. That is the reason why we need to confirm by factorizing the denominator of simplified fraction and see if any prime factor other than 2 or 5 is there or not.


(1) s is a factor of 100

Since the prime factors of 100 are only 2 and 5 . Therefore we can conclude that prime factors of s also will be either 2 or 5 or both. That means r/s will be terminating decimal.
Hence statement 1 alone is sufficient.

(2) r is a factor of 100
Here we don't have any info about the denominator s. So we cant decide whether r/s is terminating or not. Hence Statement 2 alone is insufficient.

Option A is the correct answer.

Thanks,
Clifin J Francis,
GMAT SME