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Question 26

$$a$$ is a natural number such that a has exactly two divisors and $$(a+1)$$ has exactly three divisors. The number of divisors of $$a+2$$ is


Correct Answer: 2

Solution

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Since $$a$$ has exactly two divisors, $$a$$ is prime. Since $$a+1$$ has exactly three divisors, it must be the square of a prime, say $$p^2$$, so $$a=(p-1)(p+1)$$; for this to be prime, $$p-1=1$$, giving $$p=2$$ and $$a=3$$. Then $$a+2=5$$, which is prime and has exactly 2 divisors.

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