Terminated decimal

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.

A TERMINATING decimal has a FINITE NUMBER OF DIGITS:
.5
.123
.8730253.

A NON-TERMINATING decimal has an INFINITE NUMBER OF DIGITS:
.33333....
.121212....
.871871871...

To determine whether a fraction will yield a terminating decimal:

1. Put the fraction in its MOST REDUCED FORM.
2. PRIME-FACTORIZE the denominator.

If the prime-factorization of the denominator includes ONLY 2'S AND/OR 5'S, the fraction will yield a TERMINATING decimal.
If the prime-factorization of the denominator includes ANY OTHER PRIME NUMBER, the fraction will yield a NON-TERMINATING decimal.

Case 1: 3/120
In its most reduced form, 3/120 = 1/40.
40 = 2² * 5.
Since the the prime-factorization of the denominator includes only 2's and 5's, 3/120 will yield a TERMINATING DECIMAL:
3/120 = .025

Case 2: 15/110
In its most reduced form, 15/110 = 3/22.
22 = 2*11.
Since the prime-factorization of the denominator includes 11 -- a prime number OTHER THAN 2 OR 5 -- 15/110 will yield a NON-TERMINATING DECIMAL:
15/110 = .1363636...

Onto the problem at hand.
Question rephrased: When r/s is in its most reduced form, will the prime-factorization of the denominator include only 2's and/or 5's?

Statement 1: s is a factor of 100
Since 100 = 2²5², the prime-factorization of s cannot include any prime number other than 2 and/or 5.
Thus, the value of r is IRRELEVANT:
Since the prime-factorization of s cannot include any prime number other than 2 and/or 5, the denominator of r/s in its most reduced form cannot include any prime number other than 2 and/or 5.
SUFFICIENT.

Statement 2: r is a factor of 100
No information about s.
If r=1 and s=2, then r/s = 1/2 = .5.
In this case, r/s is terminating.
If r=1 and s=3, then r/s = 1/3 = .33333...
In this case, r/s is NOT terminating.
INSUFFICIENT.

The correct answer is A.

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As you've written the question, the answer is C, as we need to know that r is an integer. (If, for example, r = 1/3 and s = any factor of 100, r/s will not terminate.)

That said, I would imagine that GMAT would tell you that r is an integer, or that r/s is expressed in reduced form (which, by convention, means that both r and s are integers), in which case the logic given above is correct. I notice that GMATGuruNY changed the prompt to add this fact to the original question, and I'd say he was right to do so.

Matt@VeritasPrep wrote:As you've written the question, the answer is C, as we need to know that r is an integer. (If, for example, r = 1/3 and s = any factor of 100, r/s will not terminate.)

That said, I would imagine that GMAT would tell you that r is an integer, or that r/s is expressed in reduced form (which, by convention, means that both r and s are integers), in which case the logic given above is correct. I notice that GMATGuruNY changed the prompt to add this fact to the original question, and I'd say he was right to do so.

Thanks! I was wondering the same.

Hey Brent
In this particular problem Statement 1 says r is a factor of 100

1,2,4,5,10,25,20,50 and 100 are factors of 100

Now let s be equal to 4 and let r be 100

Now we have (r/s)=100/25 =4

Next lets have s=10 and r=2 (r/s)=0.2(non recurring)


So I guess it should be insufficient because we could get an integer or we could also get a non recurring decimal.

Please clear this sir,

Thanks

Brent@GMATPrepNow wrote:

veenu08 wrote: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.

Thanks Sir.I was of the opinion that 4 was an integer.Do we represent that 4 as 4.0?Is this the mechanism that makes it a decimal?A little confused.Because everywhere I saw something relating to terminating decimals,it was just a 1/3 or a 25/4 and other similar decimals,never did I come across an integer.

Brent@GMATPrepNow wrote:

dddanny2006 wrote:Hey Brent
In this particular problem Statement 1 says r is a factor of 100

1,2,4,5,10,25,20,50 and 100 are factors of 100

Now let s be equal to 4 and let r be 100

Now we have (r/s)=100/25 = 4

Next lets have s=10 and r=2 (r/s)=0.2(non recurring)

So I guess it should be insufficient because we could get an integer or we could also get a non recurring decimal.

A terminating decimal is one that does not repeat.
For example, 1/4 = 0.25, so this is a terminating decimal.
Likewise, 6/3 = 2, so this is a terminating decimal
Conversely, 1/3 = 0.333333.... so this is a non-terminating decimal

Both of your examples result in terminating decimals.
100/25 = 4: 4 is a terminating decimal
2/10 = 0.2 : 0.2 is a terminating decimal

Cheers,
Brent

So you mean 100/25 equals 4.This 4 can be represented as 0.4 *10 to the power of 1.And this 0.4 is a terminated decimal.Is my understanding correct now?

Brent@GMATPrepNow wrote:

dddanny2006 wrote:Thanks Sir.I was of the opinion that 4 was an integer.Do we represent that 4 as 4.0?Is this the mechanism that makes it a decimal? A little confused. Because everywhere I saw something relating to terminating decimals, it was just a 1/3 or a 25/4 and other similar decimals,never did I come across an integer.

A decimal is just a framework one can use to represent numbers using powers of ten (this the prefix "dec"), whereby one position represents tenths, another position represents hundredths, and so on. This framework also includes tens, hundreds and thousands.

For this question, it really comes down to, "Can we write this number using decimal notation and be able to stop writing digits?"
So, if we convert 1/3 to decimal notation, we CANNOT STOP writing digits: 0.333333...
If we convert 1/4 to decimal notation, we CAN STOP writing digits: 0.25
Similarly, if we convert 12/4 to decimal notation, we CAN STOP writing digits: 3

For more, see page 110 of the OG13

Cheers,
Brent