"For each natural number m we define J_m = {0, 1, . . . , m 1}, the set of all possible remainders modulo m. Let x J₁₄₃. Define

= x mod 11, = x mod 13.

Show that x = (66 65) mod 143. (Hint: 143 = 11 13)"

Please include a detailed proof. Do not use Chinese remainder theorem. Fermats little theorem and all properties of mod are fair game.