If $p$ is a prime and $p \mid a^2$, then which must be true?

A$p \mid a$

B$p = a$

C$a$ is prime

D$p \mid 2a$ only

Answer & Solution

Correct answer: A. $p \mid a$

By the Fundamental Theorem of Arithmetic, if a prime $p$ divides $a^2$ it must appear in the prime factorisation of $a$, hence $p \mid a$. This is the lemma behind irrationality proofs.

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