XAT 2023 — QA & DI Question 12
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?
Answer & solution
- A
1
- B
3
- C
5
- D
7
None of the above
Given, x(x – p) – y(y + p) = 7p
⇒ x2 - xp - y2 - yp = 7p
⇒ x2 - y2 - xp - yp = 7pâââââââ
⇒ (x - y)(x + y) - p(x + y) = 7pâââââââ
⇒ (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).