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?
a. p^2
b. q^2
c. pq
d. p^2q^2
e. p^3q
OA - d
a. p^2
b. q^2
c. pq
d. p^2q^2
e. p^3q
OA - d
Since n is a multiple of 5, n = 5k, k being an integer.
So 5k = p^2 * q.
This means 5 either divides p or q or both.
But since p and q are primes, it means either p is 5 or q is 5 or both are 5.
Note that each of the options should always be true.
Consider a. first.
If 5 divides only q and not p, then obviously 5^2 or 25 cannot divide p^2.
Then a. will not always be true.
So a. is eliminated.
Next consider b.
If 5 divides only p and not q, then 5^2 or 25 cannot divide q^2.
Then b. will not always be true.
So b. is eliminated.
Next consider c.
Let p = 5, q = 2.
So n = p^2 * q = 50.
50 is a multiple of 5.
Here pq = 10 is not a multiple of 25.
So even c. is not always true.
Next consider d.
Now, this always has to be true, because at least one of p and q has to be 5.
This means p^2 * q^2 has to be a multiple of 25.
So d. always has to be true.
Once d. is obtained as answer, we need not check option e.
The correct answer is hence d.
