Question 103
If $k$ and $n$ are positive integers, and $k \le n$, let $M(n, k)$ denote the least common multiple of the numbers $n, n-1, \dots, n-k+1$. Let $f(n)$ be the largest positive integer $k \le n$ such that $M(n, 1) \lt M(n, 2) \lt \dots \lt M(n, k)$. Prove that:
a) $f(n) \lt 3\sqrt{n}$, for all positive integers $n$;
b) if $N$ is a positive integer, then $f(n) \gt N$ for all but finitely many positive integers $n$.
a) $f(n) \lt 3\sqrt{n}$, for all positive integers $n$;
b) if $N$ is a positive integer, then $f(n) \gt N$ for all but finitely many positive integers $n$.