just think of few no's y=2 , 3,4,5

y^2=4 , 9;16;25

so for y=2 x=2
and for y=3 x=3
for y=4 x=2
for y=5 x=5
now check for options
only D stisfy these no.

bhumika.k.shah wrote:x and y are positive integers. If x is a prime factor of y^2 , which of the following MUST be true?

A. x is even
B. x=y^2
C. y/x is even
D. x/y <=1
E. x+1 is also the prime factor of y^2

OA D

Seriously need a shortcut! else this would be tooo time consuming!

A prime factor is a prime number that could be 2 or odd in case y in this question is an even positive integer with one or few odd factors. So A is not MUST. If x is a prime factor of y^2, then y cannot be 1, hence y^2, being the square of a positive integer not 1, cannot be prime or x. Hence B is overruled. If x is a prime factor of y^2 then x is a factor of y as well, or y/x is a positive integer, not sure even or odd. Delete C. Once x is a factor of y, the maximum value that x can take, is y, hence x/y ≤ 1 is a sure remark. [spoiler]D[/spoiler]