Problem
NT-B2-M02-P013 Squares Equal to One
#13
★★★★☆ Level 4 of 5
Find all residues \(x\pmod{24}\) such that \(x^2\equiv1\pmod{24}\).
Split the condition into modulo \(8\) and modulo \(3\).
The condition \(x^2\equiv1\pmod{24}\) is equivalent to \(x^2\equiv1\pmod8\) and \(x^2\equiv1\pmod3\). Modulo \(8\), this means \(x\) is odd. Modulo \(3\), it means \(x\not\equiv0\pmod3\). Therefore, among residues \(0,\ldots,23\), the solutions are all odd residues not divisible by \(3\): \(1,5,7,11,13,17,19,23\). Direct checking confirms that each gives square \(1\pmod{24}\).
This problem already requires combining two simple tables through CRT logic.