Problem
NT-B2-M09-P017 Assigned Large Primes
Prove that for every \(k\), there exist \(k\) consecutive positive integers, each divisible by a prime divisor greater than \(k\), and these prime divisors can be chosen pairwise distinct.
Choose \(k\) distinct primes \(p_1,\ldots,p_k>k\) and impose \(N+i\equiv 0\pmod {p_i}\).
Choose distinct primes \(p_1,\ldots,p_k>k\). Consider the system \(N+i\equiv 0\pmod {p_i}\) for \(i=1,\ldots,k\), or \(N\equiv -i\pmod {p_i}\). The moduli are pairwise coprime, so by CRT there is a solution \(N_0\) modulo \(P=p_1\cdots p_k\). Every \(N=N_0+tP\) also works. For sufficiently large \(t\), \(N+i>p_i\), so each \(N+i\) is composite and has its assigned prime divisor \(p_i>k\).
This is stronger than the factorial block and better shows the flexibility of CRT.