Find all functions $f:\mathbb{N}^{+}\to \mathbb{N}^{+}$ such that (a) $f(2)=2$ (b) $f$ is [[Totally Multiplicative Functions|totally multiplicative]] (c) for all $m>n,$ $f(m)>f(n).$ ###### Solution From (b), $f(1)=f(1)^{2}$ which gives $f(1)=1$ noting that $f$ is non-zero. Suppose there exists $n$ such that for all $k\leq 2n,$ $f(k)=k.$ Then $f(2n+2)=f(2)f(n+1) = 2n+2$Now from (c), $2n=f(2n)<f(2n+1)<f(2n+2)=2n+2$which gives $f(2n+1)=2n+1.$ It follows by [[Induction Principle|induction principle]] that $f$ is indeed the identity function on $\mathbb{N}^{+}.$