Problem
NT-B2-M09-P004 A Compatible System With a Common Divisor
#4
★★☆☆☆ Level 2 of 5
Solve the system \(x\equiv 5\pmod {12}\), \(x\equiv 17\pmod {18}\).
Check compatibility, then set \(x=12t+5\).
The common divisor of \(12\) and \(18\) is \(6\), and \(17-5=12\) is divisible by \(6\), so the system is compatible. Let \(x=12t+5\). Then \(12t+5\equiv 17\pmod {18}\), so \(12t\equiv 12\pmod {18}\). Divide by \(6\): \(2t\equiv 2\pmod 3\), hence \(t\equiv 1\pmod 3\). Therefore \(x=12(3s+1)+5=36s+17\). The answer is \(x\equiv 17\pmod {36}\).
Here the period is \(\operatorname{lcm}(12,18)=36\).