Can someone please provide the solution and explanation of this modulus question

|x -9/2| = 5/2 is either x-9/2=5/2 or x-9/2=-5/2

so x=7 or 2 respectively. x can be odd or even.

since p is an odd integer and y is the median of p consecutive intergers so y can be odd or even

option 1 states that xyp is odd, now for xyp to be odd all the factors must be odd. Which is not true because x can be 2.

option 2 states that xy(p^2+p) must be even. Now for this to be even any one of the factors must be even. We know that p is a odd integer, so the square of p must be odd too. And adding p to p^2 (odd+odd) makes the sum even. Hence option 2 is correct.

(since option 2 is already proven to be correct, we don't have to check the other options, but still I am doing it for the sake of clarity, and to be sure.)

option 3 states that x^2y^2p^2 is even. So if we can prove that any one of the terms is even this too becomes true. We know p is odd. x can be 2 or 7, so we are not sure wether x is odd or even. Simmilarly y too can be odd or even depending on the number of terms. Hence we are not sure if this will be odd or even.

Hence option 2 is correct!

I hope this explanation helped you understand!

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isn't it coming (b) only?