ELMO 2020 Problem 6

For any positive integer $n$ , let

• $\tau(n)$ denote the number of positive integer divisors of $n$ ,
• $\sigma(n)$ denote the sum of the positive integer divisors of $n$ , and
• $\varphi(n)$ denote the number of positive integers less than or equal to $n$ that are relatively prime to $n$ .

Let $a,b > 1$ be integers. Brandon has a calculator with three buttons that replace the integer $n$ currently displayed with $\tau(n)$ , $\sigma(n)$ , or $\varphi(n)$ , respectively. Prove that if the calculator currently displays $a$ , then Brandon can make the calculator display $b$ after a finite (possibly empty) sequence of button presses.

Proposed by Jaedon Whyte.