NP prob: If n is a multiple of 5 and n=(p^2)q, where p and q

[sm]

Hi,

Q: 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

thanks so much in advance!

[sm]

sorry i forgot to post answer: D

i don't even know where to begin! thanks,

[San]

Hi,

Q: 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

thanks so much in advance!

if n is a multiple of 5, then n would be 5,10,15,20,25,30, 45,...100... 150,... and p and q are prime number, so p and q ---> 2,3,5,7,9,11,13....
n=(p^2)q, where n must be multiple of 5, let p=2, q=3, then n=(2^2)3=12, n is not multiple of 5. if p=2or 5, q=5 or 2, then n=(2^2)5=20, when n is multiple of 5. now you can plug in the number into the answer choice (p^2)(q^2)=(2^2)(5^2)=100 ---> it is a multiple of 25.
Note: p and q ---->one must be prime number of 5.

[Guest]

Hi,

Q: 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

thanks so much in advance!

if n is a multiple of 5, then n would be 5,10,15,20,25,30, 45,...100... 150,... and p and q are prime number, so p and q ---> 2,3,5,7,9,11,13....
n=(p^2)q, where n must be multiple of 5, let p=2, q=3, then n=(2^2)3=12, n is not multiple of 5. if p=2or 5, q=5 or 2, then n=(2^2)5=20, when n is multiple of 5. now you can plug in the number into the answer choice (p^2)(q^2)=(2^2)(5^2)=100 ---> it is a multiple of 25.
Note: p and q ---->one must be prime number of 5.

Hi San, thanks for the post! i see your reasoning but you are only proving to me why D is the correct answer :-( Is there a systemical approach to solve this problem? perhaps using a prime box? thanks,

[Guest]

Hi,

Q: 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

thanks so much in advance!

if n is a multiple of 5, then n would be 5,10,15,20,25,30, 45,...100... 150,... and p and q are prime number, so p and q ---> 2,3,5,7,9,11,13....
n=(p^2)q, where n must be multiple of 5, let p=2, q=3, then n=(2^2)3=12, n is not multiple of 5. if p=2or 5, q=5 or 2, then n=(2^2)5=20, when n is multiple of 5. now you can plug in the number into the answer choice (p^2)(q^2)=(2^2)(5^2)=100 ---> it is a multiple of 25.
Note: p and q ---->one must be prime number of 5.

Hi San, thanks for the post! i see your reasoning but you are only proving to me why D is the correct answer :-( Is there a systemical approach to solve this problem? perhaps using a prime box? thanks,

Let p=3 or 5,q=5 or 3, n=(p^2)q, where n must be multiple of 5, which of the following must be a multiple of 25?
a. P^2=3^2=9
P^2=5^2=25. so, p=9 or 25, not necessary true
b. q^2=5^2=25
q^2=3^2=9. so, q=9 or 25, not necessary true
c. pq=3*5=15 not true. because 15 is not multiple of 25.
d. (p^2)(q^2)=(3^2)(5^2)=225
(p^2)(q^2)=(5^2)(3^2)=225. true, it is a multiple of 25
e.(p^3)q=(3^3)5=45
(p^3)q=(5^3)3=375, it is not necessary true, 45 is not multiple of 25 but 375 is a multiple of 25
if you try different of prime number to p and q (one of them must be prime number of 5) such as 7,11,13...the result will be the same

so the answer is D.

[Raj]

Hello,

here is how I approached the problem:

n = 5k = p^2q

For this to be true, one of "p" or "q" has to be a 5. Now looking at the answer choices, only P^2q^2 guarantees having at least 2 5's.

A. not the answer since q could have the 5 and NOT p
B. same reason as A but p could have 5 and NOT q
C. pq could yield only one 5 and still satisfy the original equation
D.Answer
E. Same logic.. q could have the 5.

Hope this helps.
-Raj.

Hi,

Q: 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

thanks so much in advance!

if n is a multiple of 5, then n would be 5,10,15,20,25,30, 45,...100... 150,... and p and q are prime number, so p and q ---> 2,3,5,7,9,11,13....
n=(p^2)q, where n must be multiple of 5, let p=2, q=3, then n=(2^2)3=12, n is not multiple of 5. if p=2or 5, q=5 or 2, then n=(2^2)5=20, when n is multiple of 5. now you can plug in the number into the answer choice (p^2)(q^2)=(2^2)(5^2)=100 ---> it is a multiple of 25.
Note: p and q ---->one must be prime number of 5.

Hi San, thanks for the post! i see your reasoning but you are only proving to me why D is the correct answer :-( Is there a systemical approach to solve this problem? perhaps using a prime box? thanks,

[Guest]

Answer D.

p^2*q^2= n*q.
since n is a multiple of 5, then n*q is a multiple of 5, hence p^2*q^2 is a multiple of 5

[RonPurewal]

Answer D.

p^2*q^2= n*q.
since n is a multiple of 5, then n*q is a multiple of 5, hence p^2*q^2 is a multiple of 5

this is true, but it doesn't solve the problem. the problem asks for a multiple of 25, not a multiple of 5.

[RonPurewal]

a lot of the posts below contain TONS of work, to the point where they would certainly be difficult to execute within the time limit. this is not to say that you shouldn't consider such methods; on the contrary, you should consider such methods AS SOON AS POSSIBLE, if you can't think of the "textbook" theory-based approach..

here's the theory approach:

the only two primes in n's prime box are p and q.
since 5 is prime, either p or q is 5. it's impossible to tell which one.
this is actually all you need: in order to guarantee a multiple of 25, you have to square both of these primes, because you don't know which one is 5.
only (d) does this.
done.

[herjari]

I have one more doubt regarding this problem:

I know that n=(p^2)q and I have to find some way to force N to be a multiple of 25, right?

If I know that I only have prime numbers, I am certain that either p or q are 5. Therefore, I only need to square Q and I will have P and Q squared (considering that P is already squared).

Can you please tell me which part of my reasoning isn´t correct?

I thank you very much,

Hernán

[Ben Ku]

Hi Hernan,

I don't find anything wrong with your logic. The statements that either p or q is 5, and for an expression to be a multiple of 25 then both p and q should be squared are both correct. These lead to the correct answer.

Hope that helps!

[hadwhoken]

can any staff verify my solution? I got the right answer by evaluating the following:

n is a multiple of five can be rewritten as: n = 5k where K can be any number > 1.

now n = p^2 * q then:

p^2 * q = 5 * k

now p^2 /= 5 which lead to q = 5,

at this stage, we can eliminate A) C) E), left is B) D).

The reason for me to choose D) is because B) in this case is exactly 25, therefore not serving as a "multiple" of 25. Is this reasoning wrong? Did I get to the answer by luck?

[RonPurewal]

can any staff verify my solution? I got the right answer by evaluating the following:

n is a multiple of five can be rewritten as: n = 5k where K can be any number > 1.

now n = p^2 * q then:

p^2 * q = 5 * k

now p^2 /= 5 which lead to q = 5,

at this stage, we can eliminate A) C) E), left is B) D).

The reason for me to choose D) is because B) in this case is exactly 25, therefore not serving as a "multiple" of 25. Is this reasoning wrong? Did I get to the answer by luck?

you got lucky.

in the above (p^2)(q) = 5k, it's perfectly possible that p = 5 -- remember that k is NOT necessarily a prime number; it's a completely random integer, which could contain another 5.
for instance, if p = 5 and q = 3, then (p^2)(q) = 75. this is indeed a number of the form 5k, if k = 15.

and, yes, 25 is a multiple of 25.

[catennacio]

Hi all,

This is how I came up with the answer. Can anyone evaluate my reasoning?

Have a glance at all the answer choices:

A) p^2
OK, we know that n = p^2 * q is a multiple of 5, which q could involve in the making of that fact. In A, there is no information about the q, so it is not guarantied that p^2 must be a multiple of 5. Since this is a MUST BE TRUE question, we can safely conclude that p^2 is not necessarily a multiple of 5, not even 25 (5*5). p^2 can be the multiple of 25 by chances, but NOT ALWAYS (MUST BE), so eliminate A. Side note: in order for p^2 to be a multiple of 25 all the time, we need some sort of info to tell that q does NOT involve in the making of the result that p^2 * q is a multiple of 5, which is not given. Even in that case, we still have to prove a lot more, so with A, p^2 is far from the occurrence (possibility approaching 1) to be a multiple of 5, hence 25. Eliminate A.

B) q^2
For the same reasoning with A, but here since there is no info about p, so there is no guarantee (we can find lots of examples with real numbers) that q^2 is a multiple of 25. So eliminate B.

C)pq
OK, interesting. In order for any question choice to be the multiple of 25, that question choice needs to be greater than n (note that this is only a deduction, i.e. one way from left to right, not an equivalence). Since p^2 * q is a multiple of 5, and C is a taking out of one p. Since 25 > 5 and all of them (p, q, 5, 25) are positive numbers, pq is less than p^2 * q, which can't make it to be a multiple of 25, which is greater than 5. In order words, you take out one factor to make the original number less, and then you expect the resulting number to be a greater one. Impossible so eliminate C.

D)(p^2)(q^2) =(pq)^2
Hit my brain that it could be the right answer. So by reading the question stem again, I see that if n=p^2*q is a multiple of 5, which means n ends up with either 5 or 0.

If n ends with 5, then either p or q MUST BE 5 (here we can guaranty this). Since in D) (pq)^2, p and q has the same role in the formula (i.e. they role is equivalent, they have the same "weight", so when we swap them, the result of the any formulate involving p*q does not change), it is deducible that when we square up p*q, the result (pq)^2 MUST BE divisible by 25, because either p or q is 5, so (pq)^2 = 25q or 25p, hence divisible by 25. Correct answer.

If n ends with 0, it means that n is a multiple of 10 = 2 * 5, so p and q must be multiple of 5 and 2. Since again in D) pq has the same role, when we square them up either one of them, (pq)^2 will be the multiple of 25. Correct answer too.

E) (p^3)q
Now I don't have to do anymore because by the definition of GMAT answer choice, in 1 question there is only one correct answer, so I stopped here :D

But I want to try to analyze p^3*q. Since either p or q must be 5 in order for n to be divisible by 5, by looking at the asymmetric value p^3*q, we cannot guaranty that it MUST BE a multiple of 25. If p = 5 then p^3*q is always a multiple of 25, but when q = 5 then p^3*q can only be a multiple of 25 if and only if p = 5 too. In this MUST BE TRUE question, the value must always be true regardless of the conditions. In the q = 3 scenario above, we need a condition for it to be true, therefore this is not MUST BE TRUE, it could be true, if <some condition>. So, eliminate E too.

I think the core knowledge to solve this question is the symmetricity (not sure if this is written right, I'm not a native speaker) of numbers. Since 25 = 5*5, the pair of factors must be symmetric. If any answer choice is asymmetric, we can guess that it is wrong. C could be the correct answer (because its structure is symmetric), but since the question asks about 25, not 5, we need something that is greater than n to be the correct answer. So we can eliminate C.

One can make an educated guess just by looking at the answer choice as I did, because honestly I didn't know where to start, only when the abstract understanding of the making of numbers popped up, was I able to deduce the above. I couldn't solve this in my mind for 2 minutes.

Can any moderator judge whether my logic and approach is right please?

Thanks!

[RonPurewal]

catennacio --

your logic is pretty much ok, except for the part at the end about symmetry.
on the other hand, your post suggests that you may not be thinking enough -- maybe not at all, in fact -- about testing cases.

for instance, you can kill answer choice (a) pretty quickly with p = 2, q = 5; you can kill answer choice (b) quickly by reversing those values. and either of those pairs of values, in fact, is good enough to knock out choice (c).

testing cases is a bedrock skill in data sufficiency -- in fact, it may be the single most central principle in DS. you should employ it more often than it seems you currently do.

symmetricity (not sure if this is written right, I'm not a native speaker)

should be "symmetry"

of numbers. Since 25 = 5*5, the pair of factors must be symmetric. If any answer choice is asymmetric, we can guess that it is wrong.

nope.

the point of the problem has nothing to do with symmetry; the point of the problem is that either p or q -- but we don't know which -- is 5.
so, any expression in which the powers of p and q are each 2 or more will suffice. for instance, (p^2)(q^2) is a correct answer to this problem, but so are all of the following:
(p^2)(q^3)
(p^6)(q^2)
(p^9)(q^9)
(p^161109)(q^1000000)
etc.