TR-MO 1993 Problem 6

$n_{1},\ldots ,n_{k}, a$ 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. ( $x>1$ )