Question 12
Determine all pairs $(a, b)$ of positive integers for which
$$
a + b = \varphi(a) + \varphi(b) + \gcd(a, b).
$$
Here, $\varphi(n)$ is the number of integers $k \in \{1, 2, \dots, n\}$ satisfying $\gcd(n, k) = 1$.
$$
a + b = \varphi(a) + \varphi(b) + \gcd(a, b).
$$
Here, $\varphi(n)$ is the number of integers $k \in \{1, 2, \dots, n\}$ satisfying $\gcd(n, k) = 1$.