Detailed Video Solution - Weekly Quiz 2
Annotated Notes - Weekly Quiz 2 Solutions
Option B. If $p$ is a prime and $a$ is a positive integer and $p \mid a^n$, then $p^n \mid a^n$.
Solution: Suppose that $p$ is a prime and $p$ divides $a^n=a \cdot a \cdots a$. Recall that when a prime divides a product of integers then it must divide at least one of the integers contained in the product. Hence $p \mid a$. Therefore, $p k=a$ for some integer $k$. Hence, $a^n=$ $(p k)^n=p^n k^n$. Therefore $p^n \mid a^n$.
Option C:
We know $odd \times odd = odd.$
So, $odd \times odd \times odd$ is also $odd.$ (Because $odd \times odd \times odd = (odd \times odd) \times odd = (odd) \times odd = odd$ )
Similarly, Option D.
Option A is False. Counterexample: $m = 4, a=2,n=5.$ Now, we can see that $4 | 2^5$ But $4^5 $ does not divide $2^5.$
