Exercise 2.4.7. Let n be an integer. Using only the fact that every integer is even or odd, and without using Corollary 5.2.5, prove that precisely one of the following holds: either n = 4k for some integer k, or n = 4k+ 1 for some integer k, or n = 4k+2 for some integer k, or n = 4k+3 for some integer k.

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

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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: 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: For every positive integer n, prove that 7n – 3n is divisible by 4.

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Q: Find a positive integer n such that 2n is a square and 3n is a cube.

A: Sol:- Let 2n be equal to x², where x is a positive integer. Then, n can be written as n = x²/2.…

Q: Prove that the statements here are true for every positive integer n. 4 + 42 + 43 + · · · +…

A: To prove: For every positive integer, 43+44+......+4n=4(4n-16)3

Q: Let w be defined as 2 more than the number of digits in the integer w. For example, 15 =4(2 digits…

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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 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 that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

Q: Write a short proof of the following and give a counterexample If n is an integer, then 2n + 1 is…

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Q: Prove that sina de converges but not absolutely for 0 < p < 1. XP

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Q: b) Prove that () - 2) = n², where n is a positive integer.

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Q: Find all positive integers n such that 2" – 1 | 3" – 1. - -

A: Given 2n-1|3n-1 ,  for n∈ℤ+ To find if the statement is true.

Q: 1. Prove that (fn+1fn+3 – fnfn+4)* = 4 for each positive integer n.

A: We need to prove this, fn+1fn+3-fnfn+42=4

Q: Prove that 2n>n2 if n is an integer greater than 4.

A: According to the given information, it is required to prove the given statement.

Q: Negate the statement: For all positive integers, n, there exists a positive integer, k, so that…

A: The negation of a statement is:- Statement:- p implies q. Negation:- p does not imply q.

Q: . Prove that for each natural number n, /3n +2 is not a natural number.

A: Let n be any natural number.  n=3m+k2   =3m2+6m·k+k2   =9m2+6mk+k2   =33m2+2mk+k2 Here, m and k are…

Q: Prove that 2n + 3 ≤ 2n if n is an integer greater than 3.

A: Sol :- To prove:- 2n+3&lt;=2^n if n is an integer greater than 3 We prove this by induction For n=4…

Q: integers n, if n2 is odd, then n is odd. Proof: Assume that n2 is odd but n is not odd, so n must be…

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Q: Use proof by contradiction to show that the sum of two odd integers(k and l) is eve

A: An even number is a number that is divisible by two. All even numbers finish with one of the digits…

Q: Prove using PMI: For all positive integer n, 1+4+7+ ...+(3n-2) = [n (3n-1)]/2

A: The given positive integer n, To prove 1+4+7+.....+(3n-2)=n(3n-1)2 by Principal of Mathematical…

Q: Let n be any integer. Using a proof by cases, prove that n4 is either in the form of 5k or 5k+1 for…

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

A: a&gt;1, then an&gt;1