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

Question

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?

(1) s is a factor of 100
(2) r is a factor of 100

Bunuel · Accepted Answer

THEORY: Reduced fraction \frac{a}{b} (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only b (denominator) 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 250 (denominator) equals to 2*5^3 . Fraction \frac{3}{30} is also a terminating decimal, as \frac{3}{30}=\frac{1}{10} and denominator 10=2*5 . Note that if denominator already has only 2-s and/or 5-s 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 terminating decimal. (We need reducing in case when we have the prime in denominator other then 2 or 5 to see whether it could be reduced. For example fraction \frac{6}{15} has 3 as prime in denominator and we need to know if it can be reduced.) Questions testing this concept: does-the-decimal-equivalent-of-p-q-where-p-and-q-are-89566.html any-decimal-that-has-only-a-finite-number-of-nonzero-digits-101964.html if-a-b-c-d-and-e-are-integers-and-p-2-a3-b-and-q-2-c3-d5-e-is-p-q-a-terminating-decimal-125789.html 700-question-94641.html is-r-s2-is-a-terminating-decimal-91360.html pl-explain-89566.html which-of-the-following-fractions-88937.html BACK TO THE ORIGINAL QUESTION: 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? (1) s is a factor of 100. Factors of 100 are: 1, 2, 4, 5, 10, 20, 25, 50 and 100. All these numbers are of the form 2^n5^m (for example 1=2^0*5^0, 2=2^1*5^0, ...), therefore no matter what is the value of r, r/s will always will be terminating decimal. Sufficient. (2) r is a factor of 100. We need to know about the denominator. Not sufficient. Answer: A. Hope it's clear.

MacFauz · Answer

Finite decimals are decimals which end. eg 0.5,0.25, etc. Non finite are numbers like 1/3,1/6 etc ie 0.33333333333333333333333333333333333333333333333333333.......... or 0.66666666666666666666666666666666666666666666666666666..........

1)S is a factor of 100. So S cannot have more than two 2s and two 5s. Any number divisible be 2 or 5 gives a finite decimal. Since R and S are both positive integers, there can be no 0s in the decimal places either. So Sufficient

2)R is a factor of 100. Cant say anything form this. R can be 1. 1/10 is finite. 1/3 is not. Insufficient.

Answer is hence A.

IanSolo · Answer

IMO it is A, if the denominator is a factor of 100 then it could be 1; 2; 5; 10; 20.. if you divide all the positive integer by these number you will have a finite decimal result.

Sachin9 · Answer

you rock bunuel, your explanations are awesome.. thanks a lot for this one!

halloanupam · Answer

But it has been said the 'decimal with finite number of non-zero digits'. Now if we take 2/50 then it will be 0.04, which means its a finite decimal but definitely it does not have all the non-zero digits after decimal point. So, can anybody explain?

russ9 · Answer

Hi Bunuel,

Speaking from a prime perspective, I could factor "100" into prime factors and since I get 2^2 and 5^2, that should suffice in coming up with the right answer, correct? I don't need to pull up every single factors - right?

If the denominator had ANY other primes then it would NOT be a terminating decimal. Is that correct?

Bunuel · Answer

0.04 has finite number of non-zero digits: 4 is not followed by any non-zero digit.

Bunuel · Answer

Not entirely. The denominator can have some other primes as well but if those primes can be reduced the fraction still would be terminating. For example, consider fraction 3/6. The denominator has 3 in it, but it ca be reduced to get 3/6=1/2=0.5.

AndyNeedsGMAT · Answer

Why is it then 130/13 or 121/11 would give finite ... infact they properly divide...
Can someone please help ?

Bunuel · Answer

The rule above is for reduced fraction  \frac{a}{b}  (meaning that fraction is already reduced to its lowest term). When you reduce 130/13 to the lowest term you get 10 and when you reduce 121/11 you get 11: 10/(2^0*5^0) and 11/(2^0*5^0) respectively.

Check the links in my post above to practice more on this type of questions.

Ergenekon · Answer

Bunuel I did not understand from your post whether we can have other primes in denominator? Can you pls. repeat?

dabral · Answer

Ergenekon

For a fraction to be a terminating decimal the only primes that can be present in the denominator are 2 and 5, and this only applies to reduced form of the fraction. If there is any other prime in the denominator, then the fraction will be non-terminating. For example,  \frac{21}{ } ,  \frac{11}{ } , and  \frac{22}{(7^4)}  are all non-terminating decimals.

Dabral

ptanwar1 · Answer

Hello, what if r is the same as s? Then we don't have a finite decimal but an integer. Then A should not be sufficient as the answer can be a decimal or an integer. Am I wrong in my thinking?

Bunuel · Answer

An integer IS a decimal with a finite number of nonzero digits. For example, integer 51 has 2 (finite) number of digits.

kirtivardhan · Answer

Hi

In the question p/q, how can we be sure whether this can be reduced or it is already reduced

Also, if the first option would have been a multiple of 100 then the option would have been insufficient right? Since we could have any other prime factor

Harley1980 · Answer

If we have only this information, we can't say nothing about state of this fraction. 
And if we have information only about  r  or about  q  it's the same: we can't say nothing about reducing of this fraction.

I think I've got your previous question. Yes this task is exception about reducing factors:
because in this case we need to know if denominator contain any other factors except  2  and  5 . 
and we already have information that denominator is a factor of  100 , so we know that denominator doesn't conatain any factors except  2  and  5  and we don't need to reduce fraction.

Absolutely right.

oishik · Answer

''as a decimal with only a finite number of nonzero digits'' what does it exactly mean? Are 3.05, 9.002 etc under this category or sth else?

Bunuel · Answer

Yes, because there is a finite number, respectively 2 and 3, of nonzero digits after the decimal point. For example, 1/3 = 0.33333... is not a decimal with only a finite number of nonzero digits, because 3's there goes infinitely.

Goindanij · Answer

What if r=25 and s=25

Then 25/25=1

It has no decimal point in this case.

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

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

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