Problem
NT-B2-M09-P009 Squares Modulo a Composite
#9
★★★☆☆ Level 3 of 5
How many solutions does \(x^2\equiv 1\pmod {385}\) have?
Factor \(385\) into primes.
We have \(385=5\cdot 7\cdot 11\). For each of the moduli \(5,7,11\), the congruence \(x^2\equiv 1\) has two solutions: \(x\equiv 1\) and \(x\equiv -1\). The moduli are pairwise coprime, so the choices combine independently. There are \(2^3=8\) solutions modulo \(385\).
Preparation for counting solutions via CRT.