2.4 Prove that if a and b are integers and p is a prime, then (a + b)² = ap+bP (mod p). You may assume that the binomial coefficient is an integer. p! r!(p-r)!

Q: Let n be an integer. Prove that GCD(n² n + 1, n + 1) equals 1 or 3. Show by examples that both…

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Q: to show that the numbers in question are incongruent modulo n.] 11. Verify that 0, 1, 2, 22, 23,…

A: We shall solve this question in next step.

Q: Identify the expression of x for which the following equation holds for every positive integers n…

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Q: (ii) The first few cubes are 1,8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, and 1728. Write 1729…

A: The number 1729 is known as a Carmichael number. It can be represented as the sum of two cubes. For…

Q: Show that 2(p − 3)! + 1 ≡p 0, for all odd primes p.

A: Consider the given statement as follows. Show that 2p-3!+1≡0mod p, for all odd primes p. Consider an…

Q: Show that for every positive

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Q: Show that there are exactly (p−1)/2 nonzero squares mod p.

A: To prove that there are exactly p-12 nonzero squares mod p.

Q: (18)280 mod 125 =

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Q: If n| m and a = b(mod m), then a = b(mod n).

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Q: Use Pascal's Triangle to expand the binomial (x + 1)° O 25 + 5x4 + 10x + 10x2 + 5x + 1 O g5 - 5x1-…

A: Pascal's triangle…

Q: Show that if p = q + 4a (p and q are odd primes), then (p/q) = (a/q).

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Q: Show that 32n 1 is always a multiple of 8 for all n = N = {1, 2, 3,...}. We say one integer A is a…

A: Step 1:Step 2:Shows a proof by induction that 32n - 1 is always a multiple of 8 for all natural…

Q: If n is an integer, then (2n² + 3n)# 3(mod5).

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Q: Let p be a prime. Let N be the binomial coefficient (). Show that N = 2 (mod p).

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Q: (24) Let p, q € N be two distinct primes. (a) Find the following quantities (possibly in terms of p…

A: Given: p, q∈ℕ are two distinct primes. To Do: (a) Find gcdp, q,   gcdp2q, pq2,   gcdpq, p2q2+1. (b)…

Q: B. Prove this given 3+7+11+...+(4n - 1) = n(2n +1) using Mathematical induction.

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