Question 79
Original English statement:
For any positive integers $n$ and $k$, let $L(n, k)$ be the least common multiple of the $k$ consecutive integers $n, n+1, \dots, n+k-1$. Show that for any integer $b$, there exist integers $n$ and $k$ such that $L(n, k) \gt b L(n + 1, k)$.
Translation of the parallel French statement:
For any positive integers $n$ and $k$, let $L(n, k)$ be the least common multiple of the $k$ consecutive integers $n, n+1, \dots, n+k-1$. Show that for any integer $b$, there exist integers $n$ and $k$ such that $L(n, k) \gt b L(n + 1, k)$.
For any positive integers $n$ and $k$, let $L(n, k)$ be the least common multiple of the $k$ consecutive integers $n, n+1, \dots, n+k-1$. Show that for any integer $b$, there exist integers $n$ and $k$ such that $L(n, k) \gt b L(n + 1, k)$.
Translation of the parallel French statement:
For any positive integers $n$ and $k$, let $L(n, k)$ be the least common multiple of the $k$ consecutive integers $n, n+1, \dots, n+k-1$. Show that for any integer $b$, there exist integers $n$ and $k$ such that $L(n, k) \gt b L(n + 1, k)$.