How do I prove this: a) Recall that if a ∈ Z and b ∈ N, then there exist q, r ∈ Z so that a = bq + r and 0 ≤ r < b (Theorem 12.10). Prove that a ≡ r (mod b). b) Prove that r = 0 if and only if b|a.

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A: By the help of even and odd integer's we prove the result. The answer is given as follows :

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Q: Prove that if r <s < n, then P(n , s) is divisible by P(n,r).

A: Given: r≤s≤n Therefore, n-r≥n-s So we can write n-r! = n-rn-r-1n-r-2...n-sn-s-1n-s-2...1 n-r! =…

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A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

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A: Given Let m ∈ Z and consider the relation Rm on Z given by aRmb if ma-b(a) Show that Rm = R-m(b)…

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Q: rove that if r < s < n, then P(n, s) is divisible by P(n,r).

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A: Question b) not clear so only question a) answered.

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Q: 2. Z [x] is a principal ideal domain. a) True b) False O a) True O b) False

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A: Option b is correct answer

Q: ) Let F = {0, 1, a, b} be a field with 4 elements. Given that ab 1, prove that a² = b.

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A: Ideal in the ring

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A: To prove the divisibility property under the given conditions

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Q: do I prove this: Let a, b, y ∈ Z and d ∈ N so that b ≡ y(mod d). Then a ⋅ b ≡ a ⋅ y(mod d).

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