If r and s are positive integers, is r/s an integer?

Question

(1) Every factor of s is also a factor of r. 
(2) Every prime factor of s is also a prime factor of r.

Bunuel · Answer

If r and s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r. If every factor of  s  is also factor of  r , then in fraction  r/s ,  s  will just be reduced and we get an integer. Sufficient.

(2) Every prime factor of s is also a prime factor of r. The powers of prime factors of  s  could be higher than powers of prime factors of  r . eg 25/125=1/5 not an integer. Not sufficient.

Answer: A.

Hope it's clear.

galiya · Answer

i mean this one
(2) Every prime factor of s is also a prime factor of r.

Bunuel · Answer

There is an example given in my post which satisfies the given condition and doesn't give an integer value of r/s.

(2) Every prime factor of s is also a prime factor of r.  The powers of prime factors of s could be higher than powers of prime factors of r .

For example: if s=5^3 and r=5^2 then every prime of 125 (in fact its only prime 5) IS also a prime of 25 but r/s=25/125 is not an integer.

KAS1 · Answer

What if S>R? It doesn't say that R>S. It just says they are positive integers.

For first statement:
R>S = 20/10 = 2 --> Integer
S>R = 10/20 = .5 --> Not an integer

For the second statement:
R>S = 50/5 = 10 --> Integer
S>R = 5/50 = .5 --> Not an integer

That is why I picked E. Can you please explain why my reasoning is incorrect? Thanks!

Bunuel · Answer

For the first statement s cannot be greater than r. If every factor of s is also factor of r, then  r\geq{s} . Your example, (r=10 and s=20), is not possible, because 4 is a factor of s but not a factor of r.

Hope it's clear.

aeglorre · Answer

Basically, the question asks us if s is a factor of r.

1) This is significant, because we are told something about  every factor of s.  Let's say the products of the factors of s = n, then 1) gives us n*r.. is n*r/n an integer? Of course.. So 1) is sufficient

2) This only tells us a fraction of the information that 1) tells us, since 2) restricts the factors to primes.. But we don't know if s has other factors that are NOT shared by r, and thus 2) is insufficient..

So A is correct

russ9 · Answer

Hi Bunuel,

If we look at the two statements above:
(1) Every factor of s is also a factor of r. 
(2) Every prime factor of s is also a prime factor of r.

I'm having a hard time differentiating the two statements. I realize that one is talking about PF and one is talking about All Factors, but how can we assume that in statement 2, S could have it's factors raised to a higher value. Isn't the verbiage between 1 and 2 identical?

Bunuel · Answer

No, they are not identical.

(2) says that r and s have the same primes but this does not mean that r and s have the same factors. For, example, 2, 4, 8, 16, ..., 2^n all have the same prime: 2. But they certainly do not share all their factors: the factors of 8 (1, 2, 4, 8) are not the same as the factors of 4 (1, 2, 4).

Does this make sense?

petu · Answer

Hi, I answered E to this question because I thought that "every factor of s is also factor of r" does not mean that every factor of r is also a factor of s. and that is r (6) and s (4) could be an answer: every factor of s (1 and 2) are factors of r but not all factors of r are necessarily factors of s. what am i missreading? thanks!!

Bunuel · Answer

If s = 4 and r = 6, then 4, a factor of s = 4, is NOT a factor of r = 6.

AayushGMAT · Answer

r and s are positive integers.

\frac{r}{s}   is an integer?

Statement 1

Every factor of s is also a factor of r

Hence sufficient

Statement 2

every prime factor of s is also a prime factor of r

case 1 
r = 10 therefore prime factors 2 * 5
s = 10 therefore prime factors 2 * 5

\frac{r}{s} = \frac{10}{10}= integer

case 2 
r = 10 therefore prime factors 2 * 5
s= 20 therefore prime factors 2 * 5

\frac{r}{s} = \frac{10}{20}
eq{integer}

hence not sufficient

Hence Answer is A

mcelroytutoring · Answer

For condition #1, remember that this is just a fancy way of indicating that r > s.

Attached is a visual that should help.

felippemed · Answer

My suggestion: combine set properties with number properties. It becomes more intelligible. :shock:

ishan261288 · Answer

Bunuel 
statement 1 is every factor of s is also a factor of r
suppose r = 42 and s =126
factors of r: 3,2 and 7
factors of s: 3,2 and 7
Every factor of 126 that is 3, 2 and 7 is also a factor of 42 and still r/s is not integer
What am I missing here?

Bunuel · Answer

You are considering prime factors instead of factors.

The factors of 42 are: 1 | 2 | 3 | 6 | 7 | 14 | 21 | 42 (8 divisors). Prime factors of 42 are 2, 3, and 7.
The factors of 126 are: 1 | 2 | 3 | 6 | 7 | 9 | 14 | 18 | 21 | 42 | 63 | 126 (12 divisors). Prime factors of 42 are 2, 3, and 7.

As you can see not all factors of 126 are also factors of 42.

ArthurV · Answer

Why can't this sentence "The powers of prime factors of  s  could be higher than powers of prime factors of  r ." be also true for (ST1) ?

Bunuel · Answer

If powers of prime factors of  s  is higher than powers of prime factors of  r , then not every factor of s will be a factor of r. For example, if both s and r has 3 as prime and the power of 3 in s is 5 (3^5) and the power of 3 in r is 4 (3^4), then r will obviously not be a factor of 3^5, while s will be.

woohoo921 · Answer

Bunuel   avigutman 
To clarify, in statement 1 you can tell that r > s, so then you know that it will just be reduced. But in statement 2, you cannot tell whether r > s or r < s to know whether it can be reduced like it was in statement 1?

Bunuel · Answer

To be precise, from (1) it follows that r \geq s . From (2), yes, r > s, r = s, and r < s are all possible. For example: (r = 2*3 = 6) < (s = 2^2*3 = 12) ⇒ every prime factor of s (2 and 3) is also a prime factor of r ⇒ r/s ≠ integer. (r = 2*3 = 6) = (s = 2*3 = 6) ⇒ every prime factor of s (2 and 3) is also a prime factor of r ⇒ r/s = integer. (r = 2*3^2 = 18) > (s = 2*3 = 6) ⇒ every prime factor of s (2 and 3) is also a prime factor of r ⇒ r/s = integer. Hope it helps.

If r and s are positive integers, is r/s an integer?

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