Problem
NT-B1-M08-P012 Four Moduli
#12
★★☆☆☆ Level 2 of 5
Find the smallest positive \(x\) such that \(x\equiv1\pmod2\), \(x\equiv2\pmod3\), \(x\equiv3\pmod5\), \(x\equiv4\pmod7\).
First solve the first three congruences.
The first three congruences give \(x\equiv23\pmod{30}\). Check \(23,53,83,\ldots\) modulo \(7\). We have \(53\equiv4\pmod7\). Thus the smallest solution is \(53\), and all solutions are \(x\equiv53\pmod{210}\).
Stepwise merging remains convenient for four moduli.