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If \(n\) is multiple of \(5\), and \(n = p^2q\) where \(p\) and \(q\) are prime, which of the following must be a multiple of \(25\)?

A \(p^2\) B. \(q^2\) C. \(pq\) D. \(p^2q^2\) E. \(p^3q\)

\(n=5k\) and \(n=p^2p\), (\(p\) and \(q\) are primes). Q: \(25m=?\)

Well obviously either \(p\) or \(q\) is \(5\). As we are asked to determine which choice MUST be multiple of \(25\), right answer choice must have BOTH, \(p\) and \(q\) in power of 2 or higher to guarantee the divisibility by \(25\). Only D offers this.

If n is a multiple of 5 and n = p^2*q, where p and q are prime numbers which of the following must be a multiple of 25

n is a multiple of 5 and p and q are prime numbers. the only prime number which multiple of 5 i s5 itself so either p or q is 5 This is why we can surely say that p^2*q^2 is the multiple of 25 since one of thme is 5 and 5^2 = 25 so d is the answer
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n is a multiple of 5 n = pq^2 and p and q are primes numbers... so either p or q is 5 or both are 5

A. p^2 -- q could be 5, so this might not be a multiple of 25 B. q^2 -- p could be 5, so this might not be a multiple of 25 C. pq -- p could be 5 and q some other prime number,so this might not be a multiple of 25 D. p^2q^2 -- bingo, either p or q has to be 5, and this one sure will be a multiple of 25 E.p^3q - q could be 5, so this might not be a multiple of 25

Sorry (in advance) if I'm not posting this in the right place. Not sure I quite figured what to post where... Could someone help me with this question? TIA!

If n is a multiple of 5 and \(n=p^2q\), where p and q are prime numbers, which of the following must be a multiple of 25? a) \(p^2\) b) \(q^2\) c) \(pq\) d) \(p^2q^2\) e) \(p^3q\)

IMHO D

if n is a multiple of 5, it means [/m]p^2q[/m]is multiple of 5. Now both p and q are prime, so atleast one of them should be 5.

let say if p=5, then and q=3, (n=75) ,then option b is out. >>> [/m]3^2[/m] is not a multiple of 25. let say if p=3, then and q=5, (n=45) ,then option a is out. >>> [/m]3^2[/m] is not a multiple of 25. let say if p=3, then and q=5, (n=45) ,then option c is out. >>> 3 * 5 is not a multiple of 25. let say if p=3, then and q=5, (n=135) ,then option e is out. >>>[/m]3^3 * 5[/m] is not a multiple of 25.

Let see option D.

Both p or q can be 5, and if any one of them is squared, the result will be divisible by 5...!!

For this question, it's best to look at the equation and the conditions together. Here's what we know: 1. n must be a multiple of 5 2. n=p^2*q 3. p and q are prime numbers.

For n to be a multiple of 5, either p or q has to be 5. They can't be 10, 15, 25, etc. since they have to be prime numbers. As long as one of the two is 5, the other can be any prime number. Knowing this, take a look at the answer choices:

A. p^2 B. q^2 C. pq D.p^2*q^2 E.p^3*q

A and B should be eliminated, because the question asks "which of the following MUST be a multiple of 25", which means for whatever values we put in that fulfill the conditions in the stem, the correct answer choice should be 25. A and B are both at risk of either p or q being the "other" prime number (p=5 and q=3, p=3 and q=5) in which case 9 won't be divisible by 25.

C is also out-- we can finagle p and q into both being 5 to make this true, but it will not be true for every case, since p or q can just as easily be 3, and 15 won't be divisible by 25.

D is the correct answer because regardless of what p or q may be individually, the fact is that one of them will always have to be 5 and thus the result of p^2*q^2 will always be divisible by 25, which is what we're looking for in the correct answer.

E is incorrect because it's actually very similar to C, where we can potentially make it divisible by 25, but it won't be true for every case.

I hope that helps, feel free to let me know if you have any other questions!
_________________

Re: If is n is multiple of 5, and n=p^2*q where p and q are [#permalink]

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15 Jan 2012, 16:45

D is the correct answer. Tip - whenever given like n = 5k, and n = pq, always check the possibility of both p and Q as 5. I did only for 1 variable and got the answer wrong then later Bunnel post helped.

Re: If n is a multiple of 5 and n=(p^2)q, where p and q are prim [#permalink]

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19 Jan 2013, 03:17

1

This post received KUDOS

kiyo0610 wrote:

If n is a multiple of 5 and n=(p^2)q, where p and q are prime numbers, which of the following must be a multiple of 25 ?

(a)p^2 (b)q^2 (c)pq (d)(p^2)(q^2) (e)(p^3)q

either p = 5 or q = 5 Simply go to options a) p = 2 say and q = 5 b) q =2 and p = 5 c) p =2 and q =5 d) either of p or q is 5 so this will be multiple of 25 e) p = 2 and q =5

Re: If is n is multiple of 5, and n=p^2*q where p and q are prim [#permalink]

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21 Jan 2013, 07:51

In order to be a multiple of 25 we must have 5*5 derived somewhere from the equation. Also for n to be a factor of 5 the number 5 must be either p or q.

The only answer that MUST have 25 as an outcome of either p or q is D.

if n is a multiple of 5 and n=P^2q,where p and g are prime numbers, which of the following must be a multiple of 25? a. p^2 - q can be a multiple of 25

b. q^2 - - P can be a multiple of 25

c. pq - PQ is definitely a multiple of 5 but it not necessarily a multiple of 25

d. (P^2)(q^2) - Correct

e. (P^3)(q) - Assuming the worst case scenario (P is some other prime number except 5), Q is definitely a multiple of 5 but it not necessarily a multiple of 25 _________________

If you like my Question/Explanation or the contribution, Kindly appreciate by pressing KUDOS. Kudos always maximizes GMATCLUB worth-Game Theory

If you have any question regarding my post, kindly pm me or else I won't be able to reply

Re: If is n is multiple of 5, and n=p^2*q where p and q are prim [#permalink]

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27 Aug 2014, 04:39

My question is that what if n in the following question stem is equal to p^(2q), instead of (p^2)(q). how would the answer change.... My reasoning goes like this:

Since p raised to some integer power means p.p.p.p..... up to 2q. (1) and since (1) is divisible by 5, p must be divisible by 5. Hence, p^2 must be divisible by 25. Is this reasoning correct?

If \(n\) is multiple of \(5\), and \(n = p^2q\) where \(p\) and \(q\) are prime, which of the following must be a multiple of \(25\)?

A \(p^2\) B. \(q^2\) C. \(pq\) D. \(p^2q^2\) E. \(p^3q\)

My question is that what if n in the following question stem is equal to p^(2q), instead of (p^2)(q). how would the answer change.... My reasoning goes like this:

Since p raised to some integer power means p.p.p.p..... up to 2q. (1) and since (1) is divisible by 5, p must be divisible by 5. Hence, p^2 must be divisible by 25. Is this reasoning correct?

Yes, if we were told that p^(2q) is a multiple of 5 where p and q are primes, then p must be 5, which will guarantee divisibility by 25 of each option but B (q^2) and C (pq).
_________________

If is n is multiple of 5, and n=p^2*q where p and q are prime, w [#permalink]

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20 Dec 2014, 10:49

Lets say p^2 * q = 5 Then only q^2 and p^2 *q^2 can be 25 -> All other options are eliminated. Lets say p^2 * q = 25 then q^2 is eliminated. Hence D)
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Re: If is n is multiple of 5, and n=p^2*q where p and q are prim [#permalink]

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21 Dec 2015, 03:12

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If is n is multiple of 5, and n=p^2*q where p and q are prim [#permalink]

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22 Dec 2016, 03:49

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
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Re: If is n is multiple of 5, and n=p^2*q where p and q are prim [#permalink]

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26 Jan 2017, 16:47

1) Since n consists of two prime numbers p and q and it divisible by 5, either p or q have to be 5. 2) If we square both p and q, and one of them is 5, the product will have to be divisible by 5^2=25.

The correct answer is D.

gmatclubot

Re: If is n is multiple of 5, and n=p^2*q where p and q are prim
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26 Jan 2017, 16:47

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