Question 71
given positive integers $a$ and $b$ such that $a \gt b$, define a sequence of ordered pairs $(a_{l}, b_{l})$ for nonnegative integers $l$ by $a_{0} = a$, $b_{0} = b$, and $(a_{l+1}, b_{l+1}) = (b_{l}, a_{l} \bmod b_{l})$, where, for all positive integers $x$ and $y$, $x \bmod y$ is defined to be the remainder left by $x$ upon division by $y$. Define $f(a, b)$ to be the smallest positive integer $j$ such that $b_{j} = 0$. Given a positive integer $n$, define $g(n)$ to be $\max_{1 \leq k \leq n-1} f(n, k)$.
a. Given a positive integer $m$, what is the smallest positive integer $n_{m}$ such that $g(n_{m}) = m$?
b. What is the second smallest?
a. Given a positive integer $m$, what is the smallest positive integer $n_{m}$ such that $g(n_{m}) = m$?
b. What is the second smallest?