HMMT 2011 T13

Given positive integers $a$ and $b$ such that $a>b$, define a sequence of ordered pairs $\left(a_{l}, b_{l}\right)$ for nonnegative integers $l$ by $a_{0}=a, b_{0}=b$, and $\left(a_{l+1}, b_{l+1}\right)=\left(b_{l}, a_{l} \bmod b_{l}\right)$, 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) [15] Given a positive integer $m$, what is the smallest positive integer $n_{m}$ such that $g\left(n_{m}\right)=m$ ? (b) [15] What is the second smallest?