Discussion

Explanation:

Given, x(x – p) – y(y + p) = 7p

⇒ x² - xp - y² - yp = 7p
⇒ x² - y² - xp - yp = 7p​​​​​​​
⇒ (x - y)(x + y) - p(x + y) = 7p​​​​​​​
⇒ (x + y)(x - y - p) = 7p

Since, 7 and p are both prime numbers, we have

⇒ (x + y)(x - y - p) = 7 × p or 7p × 1
i.e., one (x + y) and (x - y - p) can be 7 and p or 7p and 1.

Case 1: one (x + y) and (x - y - p) can be 7 and p
⇒ (x + y) + (x - y - p) = 7 + p
⇒ 2x = 7 + 2p
⇒ x = 3.5 + 2p
This is not possible as x should be a integer.

Case 2: one (x + y) and (x - y - p) can be 7p and 1.
⇒ (x + y) + (x - y - p) = 7p + 1
⇒ 2x = 8p + 1
⇒ x = 4p + 0.5
This is not possible as x should be a integer.

∴ No value of integral value of x is possible.

Hence, option (e).

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