Problem
NT-B1-M08-P022 Prescribed Divisors of Shifts
#22
★★★★☆ Level 4 of 5
Let \(d_1,\ldots,d_k\) be pairwise coprime. Prove that there are infinitely many \(n\) such that \(d_i\mid n+i\) for all \(i=1,\ldots,k\).
Write the conditions as \(n\equiv-i\pmod{d_i}\).
The conditions \(d_i\mid n+i\) are equivalent to \(n\equiv-i\pmod{d_i}\). Since the moduli \(d_i\) are pairwise coprime, CRT gives a solution \(n_0\) modulo \(D=d_1\cdots d_k\). Then all \(n=n_0+tD\) also work. For \(t=0,1,2,\ldots\), this gives infinitely many solutions.
This is the general template for shifted-divisibility constructions.