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
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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.
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.
Rahul Lakhani
Quant Expert
Gurome, Inc.
https://www.GuroMe.com
On MBA sabbatical (at ISB) for 2011-12 - will stay active as time permits
1-800-566-4043 (USA)
+91-99201 32411 (India)
Quant Expert
Gurome, Inc.
https://www.GuroMe.com
On MBA sabbatical (at ISB) for 2011-12 - will stay active as time permits
1-800-566-4043 (USA)
+91-99201 32411 (India)