Question

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?

• A. 1
• B. 3
• C. 5
• D. 7
• E. None of the above

Answer 1

Answer: (E)

Given that \(x(x-p) - y(y+p) = 7p\), we need to find the minimum value of \(x-y\).

Let's expand and simplify the expression:

1. Expanding both terms on the left-hand side:
   
   \(x^2 - xp - y^2 - yp = 7p\)
2. Rearrange to isolate terms:
   
   \(x^2 - y^2 = xp + yp + 7p\)
3. Factorize the left-hand side as a difference of squares:
   
   \((x-y)(x+y) = p(x+y+7)\)
4. For simplicity, denote \(x-y = d\), then \(x+y = e\), giving:
   
   \(de = p(e+7)\)
5. Re-arrange to get:
   
   \(de - pe = 7p \Rightarrow e(d-p) = 7p\)
6. Since both \(x\) and \(y\) are positive integers, and since \(p\) is a prime number, \(e\) must be a multiple of \(p\).
7. The smallest positive integer solution for \(e\) in terms of \(p\) such that \(e(d-p)=7p\) is when \(e=p\). Substitute this back:
   
   \((d-p)p = 7p\)
   
   Simplifying gives:
   
   \(d-p = 7 \Rightarrow d = p+7\)

Thus, the minimum value of \(d = x-y \) is \(p + 7\). Since \(p\) is a prime number, and the smallest prime is 2, we have:

• If \(p=2\), then \(d = 2 + 7 = 9\).

Given options are 1, 3, 5, 7, and None of the above. Since none match 9, "None of the above" is correct.