If is n is multiple of 5, and n=p^2*q where p and q are prim

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 \(n=p^2q\), (\(p\) 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.

Answer: D.
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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

D

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!
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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

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...!!

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


So only D stands

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

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).
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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)

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.

n is of the form 5a where a is any positive integer.
p or q is 5, otherwise n cannot be a multiple of 5.
So to be sure that the answer is a multiple of 25, p^2q^2 is the right answer. For example, if p=5 then p^2 will be a multiple of 25 and same for q.

Can this be done with this approach, that prime factors of a perfect square will have even powers? By this theory, only D option meets the criteria. Correct me if I am wrong.

A multiple of 25 is not necessarily a perfect square. For example, 75 is a multiple of 25 but is not a perfect square.
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A common phrase that is used on the GMAT is the word must. In this question, we are asked which of the following must be a multiple of 25. This means that one of our answer choices will always be a multiple of 25, no matter what. It is our job to determine which one, based on the given information.

We are given that n is a multiple of 5, n = (p^2)q, and that p and q are prime numbers.

Because n is a multiple of 5, a prime number, we know that either p or q is 5. Let’s now analyze each answer choice to determine which one MUST (in all cases) be a multiple of 25.

A) p^2

If p = 3, then p^2 = 9 is not a multiple of 25. Answer choice A is not correct.

B) q^2

If q = 3, then q^2 = 9 is not a multiple of 25. Answer choice B is not correct.


C) pq

If p = 5 and q = 3 (or vice versa), pq = 15 is not a multiple of 25. Answer choice C is not correct.

D) (p^2)(q^2)

Regardless of which values we select for p and q, since we know that either p or q is 5, (p^2)(q^2) will always be a multiple of 25. If this is too difficult to see, let’s use numbers.

If p = 5 and q = 3, (p^2)(q^2) = (25)(9) is a multiple of 25.

If p = 3 and q = 5, (p^2)(q^2) = (9)(25) is also a multiple of 25.

Answer choice D is correct.

For practice, let’s analyze answer choice E.

E) (p^3)q

If p = 3 and q = 5, then (p^3)q = 135 is not a multiple of 25. Answer choice E is not correct.

Answer: D
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Hi All,

This question is built around a couple of Number Properties and can be solved by TESTing VALUES.

To start, we're told two things about N...
1) N is a multiple of 5
2) N = (P)(P)(Q)

Since N is a multiple of 5, at least one of it's prime factors MUST be a 5. We're told that P and Q are both PRIME, which means that P or Q or both will be a multiple of 5. This is an interesting point, since the question asks which of the following MUST be a multiple of 25 (meaning - which of these answers will ALWAYS be a multiple of 25 no matter how many different examples you can come up with?). As such, we will have to consider a couple of different possibilities...

IF...
P = 5
Q = 2
N = 50
We can eliminate answers B and C.

IF....
P = 2
Q = 5
N = 20
We can eliminate answers A and E.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

We are told n is a multiple of 5, and n = p^2*q. Since p and q are both prime, this means either p or q is 5.

Which of the following must be a multiple of 25?

A. P^2 - Doesn't have to be a multiple of 25. Could be any prime number raised to the power of 2.
B. Q^2 - Doesn't have to be a multiple of 25. Could be any prime number raised to the power of 2.
C. pq - Doesn't have to be 25. For example, 5 * 2 = 10
D. p^2 * q^2 - Correct. If we know p or q is a multiple of 5, then p^2 * q ^2 must be a multiple of 25.

Lets plug in some values to make sure. p = 5 and q = 2
p^2 * q^2 = 25 * 2 = 50

E. p^3 * q - Doesn't have to be a multiple of 25. For example p = 2 and q = 5
8 * 5 = 40

Answer is D.
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