Question 30

$$n$$ is a natural number such that $$(n + 17)$$ and $$(n - 40)$$ both are perfect square numbers, then the square-root of greatest possible value of $$(n - 40)$$ is

Correct Answer: 28

Solution

Let $$n + 17 = p^2$$ and $$n - 40 = q^2$$, so $$p^2 - q^2 = 57$$ and $$(p - q)(p + q) = 57 = 3 \times 19$$. The factor pairs $$1$$ and $$57$$ give $$p = 29$$, $$q = 28$$, while $$3$$ and $$19$$ give $$p = 11$$, $$q = 8$$. The greatest value of $$n - 40$$ is $$28^2 = 784$$, whose square root is 28.

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