Multiple of 5 and N

[GmatKiss]

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?

p^2
q^2
pq
p^2q^2
p^3q

[sanju09]

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?

p^2
q^2
pq
p^2q^2
p^3q

If n is a multiple of both 5 and n = p^2q, where p and q are prime numbers, then p^2q is also a multiple of 5, or either p or q must be 5. Only choice is [spoiler](D) p^2q^2[/spoiler] that guarantees that it's a multiple of 25.

[GmatKiss]

Not comprehensive. Could you please elaborate a bit. Thanks

[sanju09]

Not comprehensive. Could you please elaborate a bit. Thanks

If x is a multiple of 5 and x ± y is also a multiple of 5, then it means that y is also a multiple of 5. Do you agree till here?

[neelgandham]

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?

p,q are prime numbers(p!=q is my assumption).
n = p^2q and n is a multiple of 5. Implies, either p is equal to 5 or q is equal to 5. So let us check the options

A)p^2

If p=5, p^2 is a multiple of 25
If q=5, p^2 is definitely not a multiple of 25
So, option A is not the answer

B)q^2
If p=5, q^2 is definitely not a multiple of 25
If q=5, q^2 is a multiple of 25
So, option B is not the answer

C)pq
If p=5, pq is definitely not a multiple of 25
If q=5, Don't bother checking

So, option C is not the answer

D)p^2q^2

If p=5, p^2q^2 is a multiple of 25
If q=5, p^2q^2 is a multiple of 25

So, option D is the answer

E)p^3q
Don't bother checking

IMO D

[Anurag@Gurome]

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?

p^2
q^2
pq
p^2q^2
p^3q

If n is multiple of 5, and n = p²q where p and q are prime, then either p or q or both of them must be equal to 5. Let's analyze each of the cases. (Note that only one of the following can happen at a time)
1. p = 5, p² is multiple of 25, q² not
2. q = 5, q² is multiple of 25, p² not
3. p = q = 5, p² = q² = multiple of 25

We have to find a generalized expression containing p and q such that it becomes multiple of 25. From above analysis we know p² or q² is not that expression as they may or may not be a multiple of 25. But in p²q² both of them are present and simultaneously all the three cases are merged into one! For any of the above cases p²q² will be always a multiple of 25.

The correct answer is D.

[vaibhavgupta]

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?

p^2
q^2
pq
p^2q^2
p^3q

D it is!