Question 60
Let $f$ be a function defined on the set of positive integers with positive integer values.
Let's say that:
- $f$ is increasing if $n \lt m$ implies $f(n) \lt f(m)$
- $f$ is multiplicative if $\operatorname{MCD}(m, n) = 1$ implies $f(n m) = f(n) \cdot f(m)$
- $f$ is completely multiplicative if $f(n m) = f(n) \cdot f(m)$ for every $n, m$.
a. Prove that if $f$ is increasing then $f(n) \geq n$ for every $n$.
b. Prove that if $f$ is increasing, fully multiplicative and $f(2) = 2$ then $f(n) = n$ for every $n$.
c. Does statement (b) remain true if the adverb is eliminated completely?
Let's say that:
- $f$ is increasing if $n \lt m$ implies $f(n) \lt f(m)$
- $f$ is multiplicative if $\operatorname{MCD}(m, n) = 1$ implies $f(n m) = f(n) \cdot f(m)$
- $f$ is completely multiplicative if $f(n m) = f(n) \cdot f(m)$ for every $n, m$.
a. Prove that if $f$ is increasing then $f(n) \geq n$ for every $n$.
b. Prove that if $f$ is increasing, fully multiplicative and $f(2) = 2$ then $f(n) = n$ for every $n$.
c. Does statement (b) remain true if the adverb is eliminated completely?