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Question 18

The number of natural number $$n$$ for which $$n^2 + 96$$ is a perfect square is


Correct Answer: 4

Solution

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Writing $$n^2+96=k^2$$ gives $$(k-n)(k+n)=96$$, and since both factors must be even, setting $$k-n=2p$$, $$k+n=2q$$ gives $$pq=24$$. The factor pairs $$(1,24),(2,12),(3,8),(4,6)$$ each give a valid natural number $$n=q-p$$, namely $$23, 10, 5, 2$$, giving 4 values.

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