BTGModeratorVI wrote: ↑

Wed Dec 16, 2020 12:40 pm

If the integer n has exactly three positive divisors, including 1 and n, how many positive divisors does n^2 have?

(A) 4
(B) 5
(C) 6
(D) 8
(E) 9

Answer: B
Source: official guide

APPROACH #1: Start testing values in order to locate one number that has exactly 3 positive divisors
1 has 1 positive divisor (1). No good.
2 has 2 positive divisors (1 and 2) No good.
3 has 2 positive divisors (1 and 3). No good.
4 has 3 positive divisors (1, 2 and 4). GREAT!

So it could be possible that n = 4

How many positive divisors does n² have?
n² = 4² = 16
The positive divisors of 16 are as follows: 1, 2, 4, 8, 16 (5 divisors)

Answer: B
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APPROACH #2: Save a tiny bit of time by recognizing that n must be the square of an integer.
Useful properties:
If K is the square of an integer, then K will have an ODD number of positive divisors.
If K is NOT the square of an integer, then K will have an EVEN number of positive divisors.

We're told that n has exactly 3 positive divisors.
Since 3 is ODD, we know that n must be the square of an integer.
So, n COULD be 1 or 4 or 9 or 16 or 25 or.....

At this point, we need only check possible values of n that are squares of integers.
This, however, does not mean we can choose any square have an integer for the value of n. We still need to make sure that n has exactly 3 positive integers.
1 has 1 positive divisor (1). No good.
4 has 3 positive divisors (1, 2 and 4). GREAT!
From this point, our solution is exactly the same as the one covered in APPROACH #1.

Answer: B

Cheers,
Brent