If r and s are positive prime numbers, a and b are integers, and 1

Question

If r and s are positive prime numbers, a and b are integers, and 1 < a < b, then which of the following is the greatest common factor of  r^as^b  and  r^bs^a ?

A.  rs

B.  r^as^a

C.  r^bs^b

D.  r^as^b

E.  r^bs^2

Bunuel · Accepted Answer

If r and s are positive prime numbers, a and b are integers, and 1 < a < b, then which of the following is the greatest common factor of  r^as^b  and  r^bs^a ?

A.  rs

B.  r^as^a

C.  r^bs^b

D.  r^as^b

E.  r^bs^2

The GCF of two or more integers is the largest factor that all these numbers share. To find the GCF, we first perform prime factorization of the numbers. Then, we identify the common prime factors, selecting the common primes with the lowest powers. Finally, we multiply these together to obtain the GCF.

For example, the GCF of  2^3 * 3^5 * 7 * 13  and  2^2 * 3^7 * 7^4 * 11  is  2^2 * 3^5 * 7  (selecting the common primes with the lowest powers).

Applying this method, the GCF of  r^a* s^b  and  r^b*s^a  will be  r^a*s^a , once again selecting the common primes with the lowest powers.

Answer: B.

Markkawin · Answer

If r and s are positive prime numbers, a and b are integers, and 1 < a < b, then which of the following is the greatest common factor of (r^a)(s^b) and(r^b)(s^a) ?

Given that 1<a<b therefore b= a + i (i is any number. i just made it up)
Substitute b with a+i
(r^a)(s^b) = (r^a)(s^(a+i)) = (r^a)(s^a)(s^i)
(r^b)(s^a) = (r^(a+i))(s^a) = (r^a)(s^a)(r^i)
Therefore GCF is (r^a)(s^a) -> B

Posted from my mobile device

gmatophobia · Answer

We can assume values

r = 3 ;s = 7

a = 2 ; b = 4

GCF ( 3^2 * 7^4  and  3^4 * 7^2 )

We have to take the minimum power of each prime number

GCF =  3^2 * 7^2

GCF =  r^as^a

Option B

Archit3110 · Answer

prime numbers are +ve only , not sure why question says positive prime numbers?

given that r , s are prime and 1<a<b
the GCF of  r^as^b  and  r^bs^a 
will be  r^as^a

Nabneet · Answer

Silly question  gmatophobia  ,
why can't we consider r=s ?

gmatophobia · Answer

Well ! r = s, is a specialized case of the question. The answer should hold true for any generalized case. We should treat this as a 'must-be-true' question; the answer should hold true for any value of r and s.

Also the verbiage of the question indicates that we are referring to more than one number.

edthehead7 · Answer

Hi - probably an obvious answer here (I haven't spent much time going over Greatest Common Factor) but why is it known/assumed that r^a*s^a is the greatest common factor of the listed assumed values.... would love someone to just walk throught he steps they used to reach that conclusion using "GCF" ... thanks

unraveled · Answer

Although Bunuel has answered to your question I would like to add my pov(though I am nobody when its about Bunuel). Understand first what the question is asking. If r and s are positive prime numbers, a and b are integers, and 1 < a < b, then which of the following is the greatest common factor of r^as^b and r^bs^a ? A. rs B. r^as^a C. r^bs^b D. r^as^b E. r^bs^2 Here r and s are integers(it has to be prime otherwise question will be flawed) that are raised to some powers a and b that too are integers. GCF hopefully you understood as explained by Bunuel. To add to it I would say greatest common factor simply means the highest factor(prime number raised to a or b) that is present in both in r^as^b and r^bs^a . Now coming back to question, know that since a and b are not equal one accomodates other here b accomodates a. So, any number raised to power a is a small part of that number raised to power b. Hope you get the point. Looking at the answer choices A is out since it an't be highest since b > a > 1 wherein r^1s^1 would be least and we are looking for greatest one. E is out since it says indirectly(if you have understood GCF) that a = 2. This is a problem but it not a big deal since it can be so. Problem is r^b which is highest factor that can't be a common factor in both r^as^b and r^bs^a since b > a. Similarly, D is out for same reason b > a. Finally, our work becomes easy now. Among B and C, C loses out for similar reasons as we have discussed above. Note: If you know GCF then getting to answer B is easy since r and s both have to be raised to same power i.e. a that makes a part of them being raised to b. This way you can choose B straight forward. HTHs. Answer B.

KarishmaB · Answer

r and s are prime numbers so think of them as say 2 and 3. Now we know that 'a' is less than 'b' so 'b' is ('a' + something positive) say 'x.' So 'b' = 'a + x.' So what is the maximum that we can take common from: 2^a*3^{a+x} = 2^a*3^a*3^x and 2^{a+x}*3^a = 2^a*2^x*3^a ? It will be 2^a*3^a Hence, answer will be r^a*s^a . Answer (B) Check discussion on exponents here: https://youtu.be/ibDqnatAMG8

abhimanyuwhatfix · Answer

Simple, as a < b Take 'a' power common from both. (r^a.s^b , r^b.s^a) Take r^a.s^a common, again because a

satish_sahoo · Answer

Putting values can be a little time taking is what I felt, if you can see it intuitively, The GCF rule and the condition 1<a<b would help us to reach the answer quicker.  Thinking is what we need to do first .

Since, a is the smaller power, just pick "a" power of both r and s. This way it will take less than 30 secs.

So, Option B is the answer.

kaveree · Answer

still don't understand why it is not r^b and s^b? why are choosing the lower power? I read all explanations but its not making sense :/
if there is a video that explains why we choose lowest power would be great

Bunuel · Answer

Short answer would be because  r^b  is not a factor of  r^as^b  since 1 < a < b, and  s^b  is not a factor of  r^bs^a  since 1 < a < b. This is why we choose the lower powers,  r^a  and  s^a , for the greatest common factor.

noodlebob · Answer

I know the doesn't work when substituting possible values and isn't even an answer choice, but why couldn't the answer be (r^a+1)(s^a+1)?

If r and s are positive prime numbers, a and b are integers, and 1 < a

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