3.) Prove: Given an integer n, show that a. If n is even, then n - 1 is odd. b. If n - 1 is odd, then n² is even. c. Ifn? is even, then n is also even.

Q: 3. Which is a Fibonacci number? 56 189 233

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Q: 13. Recall Z = {..,-2,–1,0,1,2,..} (the integers). Let Z* = {1,2,3,.} be the positive integers. Let…

A: Given: ℤ=…, -2, -1, 0, 1, 2, … and ℤ+=1, 2, 3, …

Q: 10. If j and k are both prime numbers, and j> k, which of the following statements must be true? I.…

A: Algebra problem....

Q: Which of the following is not equivalent to the statement: If n² is even, then n+2 is even." %3D If…

A: option B is not equivalent.

Q: Q:Let S, denote the sum of the first n terms of an A.P. 3n = 6. S, = 3 S, Then prove that S.

A:

Q: Find all prime numbers up to a given integer n we proceed as follows. a. Write a number from 1 to…

A:

Q: 15%0 18.) A two-digit natural number with the same digits is equal to the sum of the positive…

A: The correct option is not available in the question.Since we want a two digit number with same…

Q: By dragging statements from the left column to the right column below, construct a valide proof of…

A:

Q: Part 4. Prove the following mathematical statements. Let n be any integer. a. If n – 1 is odd, then…

A: a. It is given that n-1 is odd.

Q: Which of the following statements are true? A. (-19)mode 5 = 9 mode 5 B. Every prime number is odd…

A: Note that (a mod b) returns the remainder after dividing a by b.So the answer can be either -4 or…

Q: i. fan is a multiple of 3. ii. fn+6 = 4fn+3 + fn.

A: I have given the proofs by mathematical induction

Q: Prove the given statement: If n is an even integer, then n2 - 1 is odd.

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Q: a. Use the formula S, = (a, + a„) to show that the sum 1, 3, 5, .. (2n – 1) = n2. b. Find the sum of…

A: a. Given formula is: Here n is the number of terms and a1 and an are the first and last terms of…

Q: (a) Suppose c is an integer. If c mod 7 = 5, what is 5c mod 7? (b) Write () in factorial form. (c)…

A: Since there are two questions posted, we will answer the first question (1). If you want the second…

Q: Let Pin) be a statement 12 +22 + 32 + . +n? = n(n +1)(2n+1) for n>0. %3D ... 6. The statement Pa) is…

A:

Q: Which of the following statements is true? a) If an integer n is odd, then n2 + 2n is even. b) If an…

A: (a) We need to know whether n2+2nis odd or even when n is odd. If n is then odd n=2k+1 for some k∈ℕ.…

Q: Prove that if 7 divides n with remainder 1, 7 divides n² + 2n + 7 with remainder 3.

A: Given, if 7 divides n with remainder 1. We have to prove that 7 divides n2+2n+7 with remainder 3.

Q: 1. For all positive integer n, which of the following expression is divisible by 6? A) 7" – 1 C) 5"…

A:

Q: For any nonzero integers a, b, c, k. One of the following statements is true A. gcd(ka, kb) =…

A: Here only two options are given to choose. Option B) and C). Also option A), D) not for choosen.

Q: Determine whether the following statements are true or false. Justify your answer with a proof or a…

A:

Q: Given any natural number ne N. Suppose that a, y> 0, then (z + y)2m > zn + 2ngn 1y 27n Proof:…

A: Statement: Given any natural number n∈ℕ. Suppose that x, y>0 then x+y2n>x2n+2nx2n-1y. Proof:…

Q: For each natural number n A, = (-→ 3+n U-1 Find UneN An and NnEN An

A:

Q: 1. Prove: If n is odd then 5n + 4 is odd.

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Q: А. If p 2q 5 is a prime number, prove that p2-q? is divisible by 3.

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Q: If n is a perfect number, prove Σ d|n 1 2.

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Q: 9. Let n be an odd number with exactly four distinct prime factors. If (3, n) = 1, prove that n is…

A: We will use knowledge of number theory to answer this question.