Let n and m be relatively prime, and let a ∈ Zn and b ∈ Zm. Define y ∗ to be the unique value in Znm such that y ∗ mod n = a and y ∗ mod m = b, whose existence is guaranteed by Theorem 7.14. Prove...

Let n and m be relatively prime, and let a ∈ Zn and b ∈ Zm. Define y ∗ to be the unique value in Znm such that y ∗ mod n = a and y ∗ mod m = b, whose existence is guaranteed by Theorem 7.14. Prove that an integer x ∈ Znm satisfies x mod n = a and x mod m = b if and only if x satisfies x mod nm = y ∗ .