Find all +ve integers

Find all integer

This is a MathCounts problem I received by email from mohit.
Find all positive integers n for which n² + 45 is a perfect square.
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Answer:

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When Is n² + 45 A Perfect Square?
Suppose n² + 45 is equal to some perfect square x². Then we have:
n² + 45 = x²
x² – n² = 45
(x + n)(x – n) = 45
We can solve the above equation in integers by considering pairs of factors of 45. The possibilities are:

45 = 1 × 45
45 = 3 × 15
45 = 5 × 9
Since x + n > x – n, we can set the sum of the numbers equal to the larger factor and the difference of the numbers equal to the smaller factor.
x + n = 45
x – n = 1
The solution is x = 23, n = 22.
x + n = 15
x – n = 3
The solution is x = 9, n = 6.
x + n = 9
x – n = 5
The solution is x = 7, n = 2.
Thus we have exactly 3 possibilities: n = 2, 6, 22.