Prove a property of divisor function
Hint: Instead of inducting on $n$ consider inducting on $k$.
Hint $\# 2$: tards answer might be a simpler way to go, but here is an alternative route:
Let $d(n)$ denote the number of divisors of the integer $n$. Show that $$ \gcd(m,n) = 1 \quad \implies \quad d(mn) = d(m)d(n). $$ Such functions are called multiplicative functions. Then just show that $$ d(p^{\alpha}) = \alpha+1. $$
One observation to help help prove that $d(mn) = d(m) d(n)$, whenever $\gcd(m,n) =1$:
If $a \mid mn$ and $\gcd(a,m) = 1$, then it must be that $a \mid n$.