Problem

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Tags: number theory unsolved, number theory



$n_{1},\ldots ,n_{k}, a$ are integers that satisfies the above conditions A)For every $i\neq j$, $(n_{i}, n_{j})=1$ B)For every $i, a^{n_{i}}\equiv 1 (mod n_{i})$ C)For every $i, X^{a-1}\equiv 0(mod n_{i})$. Prove that $a^{x}\equiv 1(mod x)$ congruence has at least $2^{k+1}-2$ solutions. ($x>1$)