Question 93
Let $\mathbb{T} = \{1, 3, 6, 10, 15, \dots\}$ be the set of triangular numbers, i.e. numbers of the form $T_n = \frac{n(n+1)}{2}$. Let $f$ be a function defined on the set of positive integers such that
1) $f(n)$ is a positive integer for each $n$;
2) $f(uv) = f(u)f(v)$ for any pair $(u, v)$ of coprime numbers;
3) $f(a + b + c) = f(a) + f(b) + f(c)$ for $a, b, c \in \mathbb{T}$.
Prove that $f(n) = n$ for all $n$.
1) $f(n)$ is a positive integer for each $n$;
2) $f(uv) = f(u)f(v)$ for any pair $(u, v)$ of coprime numbers;
3) $f(a + b + c) = f(a) + f(b) + f(c)$ for $a, b, c \in \mathbb{T}$.
Prove that $f(n) = n$ for all $n$.