Problem
NT-B2-M09-P020 A False Construction Idea
Let \(p_1,\ldots,p_k\) be distinct odd primes. Is it true that one can choose an integer \(x\) which is not congruent to \(\pm 1\) modulo any \(p_i\), but satisfies \(x^2\equiv 1\pmod {p_1p_2\cdots p_k}\)?
Check the local solutions of \(u^2\equiv 1\pmod p\).
For an odd prime \(p\), the congruence \(u^2\equiv 1\pmod p\) has only the solutions \(u\equiv \pm 1\pmod p\), because \((u-1)(u+1)\equiv 0\pmod p\). Therefore the condition “not congruent to \(\pm 1\) modulo any \(p_i\)” is incompatible with \(x^2\equiv 1\pmod {p_i}\) for every odd prime. Such an \(x\) does not exist for any choice of \(p_i\). Answer: the claim is false.
This is a deliberate trap: CRT does not override local restrictions. Students should be able to reject a false construction idea.