I need some help with the following question, thanks.
Let $$r : \mathbb N → \mathbb N$$ be the function where the output is the number of positive integer factors of the input.
For example, $$r (15) = 4\ \ (4\ \ positive\ \ integer\ \ factors\ \ of\ \ 15)$$
Questions I am confused about:
$1)$ Is $r$ an injective function$?$
$2)$ If $n = p^k$, where $p$ is prime, and $r$ is a positive integer, find a formula for $r (n)$ .
$1)$ . $r$ is not injective . Since every prime number has $2$ positive integer divisors. $$r(p)=2$$ for any prime $p$.
$2)$ . $$n=p^k$$ Then the divisors of $n$ are $$1=p^0,p=p^1,p^2,.....,p^{k-1},p^k$$
So you see , there are $k+1$ such divisors .
So,$$r(p^k)=k+1$$