Problem
NT-B1-M12-P024 Set 6. Consecutive Numbers with a Square Divisor
#24
★★★★★ Level 5 of 5
Prove that for every \(k\ge1\), there exist \(k\) consecutive positive integers, each divisible by the square of some prime.
Choose distinct primes \(p_1,\ldots,p_k\) and impose \(n+i\equiv0\pmod{p_i^2}\).
Choose distinct primes \(p_1,\ldots,p_k\). Consider the system \(n\equiv-1\pmod{p_1^2}\), \(n\equiv-2\pmod{p_2^2}\), \(\ldots\), \(n\equiv-k\pmod{p_k^2}\). The moduli \(p_i^2\) are pairwise coprime, so CRT gives a solution \(n_0\). Then each \(n_0+i\) is divisible by \(p_i^2\). Adding a sufficiently large multiple of \(p_1^2\cdots p_k^2\) to \(n_0\), we obtain positive consecutive integers.
Strong final construction for the first level.