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!

sorry i forgot to post answer: D

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

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.

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.

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.

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.