5. Use direct proof technique to prove the following theorem: "If n is an odd integer, then n³ + 1 is an even integer."

Q: Prove that if a is an odd integer, then 5a² – 3a + 5 is odd.

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Q: Prove n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

A: We have to proff the question

Q: Prove or disprove "If 2m + 1 is an odd prime number, then m = 2n for some nonnegative integer n."

A: Given if  is an odd prime number, then  for some nonnegative integer n.The aim is to prove or…

Q: Prove that the statement is true for every positive integer n.

A: -11+-12+-13+---+-1n=-1n-12. We prove this statement by using the Principle of Mathematical…

Q: Give a direct proof of the theorem “The sum of any two odd integers is even”.

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Q: Prove that 1+2+3+4+...+n= n² + n 2 for every positive integer

A: We have to prove the statement using the principle of mathematical induction.We know thatPrincipal…

Q: 1. Prove that nº + n is even for every integer n.

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Q: Claim: For every odd integer n, n² - 1 is a multiple of 8. Purported proof: Suppose there is an…

A: The given claim is "For every odd integer n, n2−1 is a multiple of 8".Purported proof:Suppose there…

Q: Prove. A number is triangular if and only if it is of the form (n(n+1))/2 for some nonnegative…

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Q: Find all integers 100 ≤ n ≤ 999 such that the last three digits of n² is same as n.

A: To find: All integers  such that the last three digits of is same as n.

Q: Prove or disprove For every integer m, m(m+1) is even

A: To prove or disprove the given statement- For every integer m , m(m+1) is even .

Q: (2) Use a proof by ca. If is int er

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Q: Determine which of the following “proofs” are correct and which are incorrect. If a proof is…

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Q: A. Prove that for all positive natural numbers n, 13 + 23 + . + n³ = n²-(n+1)²/4. + ...

A: Explanation of the answer is as follows

Q: 1. Show there exists an integer n such that n² + 3n/2 = 1. Is the integer unique? Explain your…

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Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: Prove that 2n+1 +(-1)" 3-2 whenever n is a nonnegative integer.

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Q: Consider the two statements: P: For every integer n there exists an integer m such that m + n = 0.…

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Q: Prove that p=5 and q=2 are the only pair of prime numbers such that p−q=3.

A: you can solve it by using the solution of Diophantine equation. So , Here we have to prove  p = 5…

Q: Prove that n²– 3n + 1 is odd for every integer n.

A: To prove that n^2-3n+1 is odd for every integer n.

Q: Prove the following statement by contradiction: If (n – 3)2 is an odd integer, then n is an even…

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Q: k a] lim = 0 kER %3D

A: 2(a) In this question, we prove the limit of the given sequence by using   ε-δ definition.

Q: Use cases to prove that for all integer n, n2+3n+5 is odd.

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Q: (7) The difference of two odd integers x and y is 4. Prove that x and y are relatively prime.

A: Given difference of two odd integers is 4.

Q: 20. (a) Fill in the blank: 1+3+5+..+ (2n – 1) = k=1 (b) Use part (a) to prove that the sum of the…

A: This is an arithmetic series with common difference (3-1)=2 And this has n terms in it So we use the…

Q: 4. Find all the prime numbers that can be expressed in the form n2 – 9 for some integer n.

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Q: Prove that 3 | 22n+1 +1 for all natural numbers n.

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

Q: 17. Prove by induction that 1³+23++n³= (1+2+...+n)² for all positive integers n, that is, the sum of…

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Q: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p

Q: Let m be an integer, if m is an even integer then prove that m + 7 is odd

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Q: (2) Prove the statement "Every positive even integer less than 16 can be written as a sum of three…

A: Perfect square: n is a perfect square, if and only if, there exist an integer k such that n=k2.…