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.
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