Problem
NT-B1-M11-P019 Three Conditions on Consecutive Numbers
#19
★★★☆☆ Level 3 of 5
Find one positive integer \(n\) such that \(2\mid n+1\), \(3\mid n+2\), and \(5\mid n+3\).
Write the conditions as congruences for \(n\).
We need \(n\equiv1\pmod2\), \(n\equiv1\pmod3\), \(n\equiv2\pmod5\). The first two give \(n\equiv1\pmod6\). The numbers are \(1,7,13,19,\ldots\); residue \(2\) modulo \(5\) gives \(7\). Thus \(n=7\) works.
Gentle transition to CRT constructions.