Problem
NT-B1-M11-P024 A Long Block of Composite Numbers
#24
★★★★★ Level 5 of 5
Prove that for every \(k\ge1\), there exist \(k\) consecutive positive integers, each composite.
Try a number divisible by all numbers \(2,3,\ldots,k+1\).
Take \(N=(k+1)!\). Then \(N+2,N+3,\ldots,N+k+1\) are \(k\) consecutive numbers. For each \(i=2,3,\ldots,k+1\), the number \(N+i\) is divisible by \(i\), because \(N\) is divisible by \(i\). Also \(N+i>i\), so \(i\) is a nontrivial divisor. Therefore all these numbers are composite.
Classic construction, very important for olympiad thinking.