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## 01. Increasing multiplicative functions

Happy new year, everyone! In this week, we propose two problems (the first one is easy and the second one is challenging):

01a. PEN K12 (Canada 1969) Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for all
$m, \, n\in \mathbb{N}$: $f(2) = 2$, $f(mn) = f(m)f(n)$, $f(n + 1) > f(n)$.

01b. Let $f:\mathbb{N} \to \mathbb{R}^{+}$ be a function satisfying the conditions:
(a) $f(mn) = f(m)f(n)$ for all relatively prime $m$ and $n$, and
(b) $f(n+1) \geq f(n)$ for all positive integers $n$.
Show that there is a constant $\alpha \in \mathbb{R}$ such that $f(n)=n^{\alpha}$ for all $n \in \mathbb{N}$.

* Any comments are welcome! You can also disscuss the problems here!
* To get the current edition of PEN Problems Book, visit here!
* Next week, the solutions will be uploaded here in the pdf file.