Discrete Math Proof: Mod Properties

"For each natural number *m *we define *J _{m}*

*= {0, 1, . . . , m − 1}*, the set of all possible remainders modulo m. Let x ∈ J

_{143}. Define

α = x mod 11, β = x mod 13.

**Show that x = (66β − 65α) mod 143. (Hint: 143 = 11 · 13)**"

How can this proof be shown using mod properties and/or Fermat's Little Theorem?

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