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Prime number $a$ can be divided by $3$ with no remainder. Which of the following is a prime number?

$$A) a^{2} + a$$ $$B) 2a + 4$$ $$C) a^{2} + 2$$ $$D) a^{3} - 2a^{2} - 2$$ $$E) 4a + 3$$

I assumed $a$ to be $3$ since if it can be divided by $3$ but it isn't $3$, then it can be divided by a number other than $1$ and itself which violates the condition to be a prime number. Option $C$ resulted in $11$ and option $D$ resulted in $7$ while others didn't result in prime numbers. Answer key says $D$ is correct however so is $C$ if we accept $a$ to be $3$. Is my assumption that $a$ is $3$ wrong? If it is, would you please help me solve it?

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    Prime number $a$ is divided by $3$ with no remainder $\implies a=3$. So yes, there are two correct answers here. – 2017-01-05
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    I concur with your reasoning. Perhaps the question has a typo in it. – 2017-01-05
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    @barakmanos That's what I figured as stated. However, try it with both option $C$ and $D$ and you'll see that it seems to have two correct answers. I assume it is a mistake on the publishers side then? – 2017-01-05
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    This does seem to be a mistake in the formulation or printing of the problem in the book. Perhaps there was an accidental substitution in choice C, or perhaps the person who wrote the problem made an arithmetic error. It does happen sometimes. – 2017-01-05
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    Yes but the mistake could be (for example) that they forgot to mention that both C and D are correct answers. You are asking to speculate exactly what is the mistake on the question, and the answer to that can only be... well, speculated... – 2017-01-05
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    Nothing is said about the sign of $a$. Depending on the context, it might be also $-3$, but both options C and D yield prime numbers in this case, too. – 2017-01-05
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    @ajotatxe: The term *prime number* typically refers to natural numbers only (otherwise there are no prime numbers except for minus $1$). – 2017-01-05
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    So the correct answer is C and D. We get so used to certain rules in the presentation of test questions that we make unsubstantiated assumptions. Have you ever seen a true-false test that read, "Write T before each statement that is true; write F before each statement that is false, and write only one letter before each statement? – 2017-01-05
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    @Airymouse I sure did. However, this isn't that type of test. I'm convinced that it is a typo. – 2017-01-05

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