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.

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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Margarette
Magoosh Test Prep

fro me its D P^2*q^2

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.

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

Merging similar topic. Please refer to the solutions above.
_________________

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. NEW!!!

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set. NEW!!!


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well i had the same thing in mind..so i wont i have to write it i guess

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