Let $L(n)$ denote the least common multiple of $\{1, 2 . . . , n\}$. (i) Prove that there exists a positive integer $k$ such that $L(k) = L(k + 1) = ... = L(k + 2000)$. (ii) Find all $m$ such that $L(m + i) \ne L(m + i + 1)$ for all $i = 0, 1, 2$.
Source: 2001 Singapore TST 2.3
Tags: LCM, least common multiple, number theory
Let $L(n)$ denote the least common multiple of $\{1, 2 . . . , n\}$. (i) Prove that there exists a positive integer $k$ such that $L(k) = L(k + 1) = ... = L(k + 2000)$. (ii) Find all $m$ such that $L(m + i) \ne L(m + i + 1)$ for all $i = 0, 1, 2$.