the number of solutions of the equations

Dear gaurav_raos,

I'm happy to respond.

My friend, this is DEFINITELY NOT a math question of GMAT quality. It has a few problems.

First of all, it is not clear at all whether the author intended m and n to be restricted to the integers. Those letters are often used for integers, and restricting it to integers would make the problem more interesting mathematically, but no restriction is given.

In the absence of such a restriction, there must be an infinitive number of solutions. The first few are
\(n = \pm1\), \(m = \pm\sqrt{1615}\)
\(n = \pm\sqrt{2}\), \(m = \pm\sqrt{1616}\)
\(n = \pm\sqrt{3}\), \(m = \pm\sqrt{1617}\)
\(n = \pm2\), \(m = \pm\sqrt{1618}\)
\(n = \pm\sqrt{5}\), \(m = \pm\sqrt{1619}\)
etc. Those are just some. The variable n could take on decimal values, multiple of pi, etc. etc. and for each value, positive or negative, there would be a positive or negative value of m. There is a continuous infinitive of possible solutions. Answer = (E).

Perhaps that version, the non-integer version, was intended as the solution, but that's not so interesting because of the infinite panoply of possible solutions.

If we add the stipulation that m & n are integers, that's a much more interesting question, but one that exceeds the difficulty of the GMAT. Rewrite the equation as:
\(m^2 - n^2 = 1614\)
Here's some basic number theory that is beyond what you need to know for the GMAT.
1) The difference between two consecutive squares is always an odd number. Any odd number greater than 1 is a difference of the squares of consecutive integers.
2) The difference between \(n^2\) and \((n + 2)^2\) is always divisible by 4.
3) The difference between \(n^2\) and \((n + 3)^2\) is always divisible by 3. It's always 3 times an odd number.
4) The difference between \(n^2\) and \((n + 4)^2\) is always divisible by 8.
5) The difference between \(n^2\) and \((n + 5)^2\) is always divisible by 5. It's always 5 times an odd number.
6) The difference between \(n^2\) and \((n + 6)^2\) is always divisible by 12.
etc.

The number 1614 is not odd and not divisible by 4. It is divisible by 3, but it's three times an even number. The prime factorization is:
1614 = 2*3*269
From these patterns about the difference between squares of integers, we see that 1614 cannot be the difference between the squares of any pair of integers. Answer = (D). Once again, that's a much more interesting question mathematically, but it's 100% beyond what the GMAT would expect you to know.

This leads to an even more interesting question:
For the integers P and Q, how many possible solutions exist for the equation \(P^2 - Q^2 = 1155\)?
Since that number, 1155 = 3*5*7*11, there are potentially a host of possible solutions. Of course, this is way beyond the GMAT.

Does all this make sense?
Mike