tdkk123 wrote:If n is a positive integer and is not the square of any integer, is it a prime number?
(1) Among the factors of n, only n is greater than √n
(2) among the factors of n, only 1 is less than √n

OA D


Totally stumped!! Can someone explain?

If you take a positive integer x which is not a perfect square, then if x is *not* prime, it must be possible to write x as a product of two smaller integers:

x = a*b

If x is not a perfect square, then a and b must be *different* (and if a and b are both less than 24, neither of them is equal to 1). So, for example, if x were 24, we could write x as a product of two smaller integers in several ways (2*12, 3*8, 4*6). Now, if x = a*b, then one of the numbers a or b must be smaller than √x, and the other must be larger than √x. They can't both be smaller than √x, because then their product would be smaller than √x*√x = x, and they similarly can't both be larger than √x, because then their product would be greater than x. In the specific example of 24, you can see that in each of the three products (2*12, 3*8, 4*6) one of the factors is less than √24 ≈ 5, and one is greater.

So if x is not a prime, and not the square of a prime, x must have at least one factor which is somewhere strictly between 1 and √x, and another which is strictly between √x and x. That makes the answer D here.