If $p$ is a prime number and $p\mid a^n$ (where $a$ is an integer and $n$ is a positive integer), then which of the following must be true?

Step 1 (Euclid's Lemma): A fundamental theorem in number theory (Euclid's Lemma) states that if a prime number $p$ divides a product of two integers $x\cdot y$, then $p$ must divide $x$ or $p$ must divide $y$.

Step 2 (Apply to prime powers): We can express $a^n$ as a product of $n$ factors of $a$:

If $p\mid a^n$, we apply Euclid's Lemma repeatedly:

• $p\mid (a\cdot a^{n-1})⟹p\mid a$ or $p\mid a^{n-1}$.

• If $p\mid a$, we are done. If $p\mid a^{n-1}$, we repeat the split.

By mathematical induction on $n$, the prime $p$ must divide the base $a$:

Step 3 (Conclusion): The relation $p\mid a$ must be true. This is at option index 0.