Problem
NT-B1-M08-P019 Residues Modulo \(5,8,9\)
#19
★★★☆☆ Level 3 of 5
Find the smallest positive \(x\) if \(x\equiv1\pmod5\), \(x\equiv3\pmod8\), \(x\equiv4\pmod9\).
First combine moduli \(5\) and \(8\).
From \(x\equiv3\pmod8\): \(3,11,19,27,35,43,\ldots\). Residue \(1\) modulo \(5\) first occurs at \(11\), so \(x\equiv11\pmod{40}\). Now \(11,51,91,131,171,211,\ldots\); the first with residue \(4\) modulo \(9\) is \(211\). Thus \(x\equiv211\pmod{360}\).
Teaches students not to fear modulus \(360\).