Integers: Even and Odd - Rules of Addition and Subtraction

If $$p$$ and $$q$$ are distinct prime numbers, and $$n = pq + p + q$$, which of the following CANNOT be true?

Incorrect. [[snippet]] If you pick $$p = 2$$ and $$q = 3$$, then $$n = 2 \cdot 3 + 2 + 3 = 11$$, which is odd. So this isn't the correct answer.

Quite right. There's only one even prime number, 2. If $$p$$ (or $$q$$) is equal to 2, then $$n = \mbox{Even} + \mbox{Even} + \mbox{Odd} = \mbox{Odd}$$. Otherwise, both $$p$$ and $$q$$ are odd, and $$n = \mbox{Odd} + \mbox{Odd} + \mbox{Odd} = \mbox{Odd}$$. So $$n$$ will always be odd, meaning it can never be even.

Incorrect. [[snippet]] If you pick $$p = 2$$ and $$q = 3$$, then $$n = 2 \cdot 3 + 2 + 3 = 11$$, which is prime. So this isn't the correct answer.

Incorrect. [[snippet]] This is a difficult answer to find a counterexample for. If you pick $$p = 2$$ and $$q = 11$$, then $$n = 2 \cdot 11 + 2 + 11 = 35$$. Since this is divisible by 5, it's not the correct answer.

Incorrect. There's not a "largest" prime number. There's a prime number larger than 9,999,999, so if you choose that prime number for $$p$$, then $$n$$ will be larger than 9,999,999. [[snippet]]

$$n$$ is odd.

$$n$$ is even.

$$n$$ is prime.

$$n$$ is divisible by 5.

$$n$$ is greater than 9,999,999.