Problem
NT-B1-M08-P006 Three Moduli
#6
★★☆☆☆ Level 2 of 5
Find the smallest positive \(x\) such that \(x\equiv1\pmod3\), \(x\equiv2\pmod4\), \(x\equiv3\pmod5\).
First combine the conditions modulo \(3\) and \(4\).
From \(x\equiv2\pmod4\), check \(2,6,10,14,\ldots\). Residue \(1\) modulo \(3\) first occurs at \(10\), so \(x\equiv10\pmod{12}\). Now \(10,22,34,46,58,\ldots\); the first with residue \(3\) modulo \(5\) is \(58\). Thus \(x\equiv58\pmod{60}\).
Shows step-by-step merging of congruences.