Problem
NT-B1-M08-P020 Residues Modulo \(7,9,11\)
#20
★★★☆☆ Level 3 of 5
Find the smallest positive \(x\) such that \(x\equiv2\pmod7\), \(x\equiv5\pmod9\), \(x\equiv8\pmod{11}\).
Merge the first two congruences, then the third.
The numbers \(2,9,16,23,30,37,44,51,58,\ldots\) have residue \(5\) modulo \(9\) at \(x=23\). Thus \(x\equiv23\pmod{63}\). Check \(23,86,149,212,275,338,\ldots\); \(338\equiv8\pmod{11}\). The answer is \(x\equiv338\pmod{693}\).
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