Question 8
A function $f$ from the positive integers to the positive integers is called Canadian if it satisfies
$$
\operatorname{gcd}(f(f(x)), f(x+y)) = \operatorname{gcd}(x, y)
$$
for all pairs of positive integers $x$ and $y$.
Find all positive integers $m$ such that $f(m) = m$ for all Canadian functions $f$.
$$
\operatorname{gcd}(f(f(x)), f(x+y)) = \operatorname{gcd}(x, y)
$$
for all pairs of positive integers $x$ and $y$.
Find all positive integers $m$ such that $f(m) = m$ for all Canadian functions $f$.