Problem
NT-B1-M08-P023 A Block with Prescribed Prime Divisors
#23
★★★★☆ Level 4 of 5
Prove that there exist \(6\) consecutive positive integers, each divisible respectively by one of \(5,7,11,13,17,19\).
Look for \(n\) such that \(n+i\) is divisible by the corresponding prime.
We need to solve \(n\equiv-1\pmod5\), \(n\equiv-2\pmod7\), \(n\equiv-3\pmod{11}\), \(n\equiv-4\pmod{13}\), \(n\equiv-5\pmod{17}\), \(n\equiv-6\pmod{19}\). The moduli are pairwise coprime, so CRT gives a solution. For any such \(n\), the numbers \(n+1,\ldots,n+6\) have the required divisors. Taking a sufficiently large positive solution makes them positive and larger than their divisors.
The student should see that an explicit number is not always needed.