Let x and y be two positive integers and p be a prime number. If x (x - p) - y (y + p) = 7p, what will be the minimum value of x - y?

The given equation is,

x (x - p) - y (y + p) = 7p

$$x^2-px-y^2-py=7p$$

$$x^2-y^2-px-py=7p$$

$$\left(x+y\right)\left(x-y\right)-p\left(x+y\right)=7p$$

$$\left(x-y-p\right)\left(x+y\right)=7p$$

As '7' & 'p' both are prime numbers

$$\left(x-y-p\right)\left(x+y\right)$$ can be expressed as $$\left(7\times\ p\right)\ or\ \left(7p\times\ 1\right)$$

Case (i) - $$\left(x+y\right)\ \times\ \left(x-y-p\right)=7\times\ p$$

                $$x+y+x-y-p=7+ p$$

                $$2x-p=7+p$$

                $$x=\frac{7}{2}+p$$

But it's given that 'x' is a positive integer. This case is not possible.

Case (ii) - $$\left(x+y\right)\ \times\ \left(x-y-p\right)=7p\times\ 1$$

                 $$x+y+x-y-p=7p+1$$

                 $$2x-p=7p+1$$   

                 $$x=\frac{1}{2}+4p$$

But it's given that 'x' is a positive integer. This case is not possible.

The given equation is not possible with given conditions.

Option (E) is correct.