85. (a) Prove that any integer is expressible in the form 4n, 4n + 1, 4n + 2, or 4n + 3 where n is an integer.

Q: 2.) Let 1 > -1 be a real number. Prove that (1+1)" >1+ nx for all integers n > 1.

A: We use principal of mathematical induction (PMI) to prove this statement.

Q: Prove that 1– 2+2² – 2° + -.-+(-1)"2"=2**1 for all nonnegative integers n.

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Q: O Prove 1+3+ 9+27+ ·+ 3n-1 = 3"-1 for any positive integer n. %3D

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Q: Prove that the analogue of lemma 1.23 is not true for numbers of the form 4n+3 where n is an…

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Q: n^2 +1 is not divisible by 3 for every integer n.

A: Any integer n can be written in the form of: n=3a+b, where b = 0 or 1 or 2 Case 1: If b=0, n2+1=…

Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: 3n+2 and 5n+3 are relatively prime for every integer n.

Q: Prove 3 | 17k+1-2*5k+3+11k+3 for k>=0 (nonnegative integer)

A: Here in this problem we have to prove that for k≥0 i.e k is a non negative integer…

Q: 1.) Prove every integer n >= 1, 1+5+9...+(4n-3) = n(2n-1).

A: 1.Apply mathematical induction to prove the statement. Let the given statement be Pn:…

Q: Prove that a number in the form 6k+5,” what I mean is a number in the form of 6 times some integer…

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Q: 8"-3 is a multiple of 5. In other words, prove that 8"-3"=5 for some i integer j.

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Q: The sum of any two integers of the from 4k + 1is always: prime Zero Od Even

A: Given statement, The sum of any two integers of the form 4k + 1 is always……

Q: 2n 10. Prove that 31 (5 - 1) for every integer n 2 0.

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Q: Let b a natural number and q1, q2, r1, r2 integers with 0 ≤ r1 < b and 0 ≤ r2 < b such that q1b + r1…

A: Natural Numbers: Natural numbers, that embody all positive integers from one to time, square measure…

Q: 1. Prove that the negative of any even integer is even. 2. Prove that the difference between an even…

A: "Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: 75. Prove that the sum of an even integer and an odd integeris odd.

A: sum of an even integer and an odd integer is odd.

Q: rove that the difference of any two odd integers is even.

A: Even numbers are of the form 2k. Odd numbers are of the form 2k+1, where k is an integer.  Let a and…

Q: Show that there are infinitely many primes to prove that there are infinitely many primes of the…

A: This question is related to Number Theory. We solve the result by contradiction.

Q: Abstract Algebra

A: To find non-negative representations under the given conditions

Q: 7) Use a direct proof to show that: a) The sum of an even integer and an odd integer is odd. b) If n…

A: Use the following concepts, to obtain the required result. An even integer m can be written as a…

Q: Prove or disprove: 2n + 1 is prime for all nonnegative integers n.

A: Prove or disprove: 2n + 1 is prime for all nonnegative integers n.

Q: Suppose n₁, n₂ are positive integers and gcd(01, n₂) = 1. Then for every integer x, if n₁x and n₂|x,…

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Q: 1. Prove that Si-2' = (n – 1) - 2"*+1 + 2 for any positive integer n.

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Q: 5. Prove that the integer 3". 22" – 1 is divisible by 11 for all n > 0.

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Q: 32. Prove that 3 divides n + 2n whenever n is a positive integer.

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Q: Prove that n- + 1> 2" when n is a positive integer with 1 <n<4.

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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: 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: Q1 Use mathematical induction to prove that (-5)"+1 – 1 3 + 3 * (-5) + 3 * (-5)² + 3 * (-5)³ + .… +…

A: Given equation is, 3+3(-5)+3(-5)2+3(-5)3+......+3(-5)n=(-5)n+1-1-2

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: Let 2k +1 be a prime number. Prove that then k = 0 or k = 2" for some n > 0. %3D

A: Solution:- Let 2k+1 be a prime number k=0 or k=2n for some n≥0   For, k=0 2k+1=20+1 2k+1=1+1…

Q: Show that the product of two numbers of the form 4n + 1 is still of that form. Hence show that there…

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Q: Prove that 2 - 2 · 7 + 2 · 72 – ... + 2(-7)" = (1 – (-7)n+1)/4 whenever n is a nonnegative integer.

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Q: Prove: Any positive integer can be represented as an aggregate of different powers of 3, the terms…

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Q: how many positive integers n satisfy 22-5n>-3?

A: 22-5n&gt;-3

Q: Prove that 11 is the only prime number of the form 4n2 - 25.

A:

Q: Prove that 1 + 3 + • • • + (2n - 1) = n2 for all natural numbers n.

A: Our Aim is to prove that 1 + 3 + .... + (2n - 1) = n2 for all natural numbers n.

Q: 1. Prove that 3 + 3 x 5 + 3 x 52 + 3 x 53 + ..+ 3 x 5° = 3(5n+1 – 1)/4 whenever n is a nonnegative…

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Q: Prove that a natural number (an-1an-2a1ao)10 is divisible by 9 if and only if the sum of the digits…

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Q: (b) Prove that 2m ≤ m! + 2 for m ¤ N. (c) For any nonzero real number x, prove ---3 i=1 = 1- x -n

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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: (a) Show that 1++++ 1 21+, 2n-1 2n whenever n is a nonnegative integer.

A: We need to show that 1+12 + 13 + ..... + 12n-1 + 12n ≥ 1+n2We will show this by mathematical…

Q: Let's take an integer n to be odd if there is some integer k such that n = 2k – 1. Prove that the…

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Q: Prove 8^ +l is composic for any poSitive integer n.

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Q: Determine whether the statement is true or false. Justify your answer with a proof counterexample,…

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Q: prove that n^2 +1 >= 2^n when is a positive integer with 1<=n<=4

A: To prove n^2 +1 &gt;= 2^n when is a positive integer with 1&lt;=n&lt;=4

Q: Prove: 5" +9 < 6" for all integers n2 2.

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Q: Prove that any positive integer N is divisible by 11 if and only if the difference between the sum…

A: Consider the provided question, we need to prove that, Any positive integer N is divisible by 11 if…

Q: Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4. In…

A: Introduction: Divisibility is a property of an integer. For two integers a and b with b&gt;a, the…

Q: 3. Prove that 0 divides an integer a if and only if a 0.(In your proof. you can freely ILNE your…

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Q: oof. Assume that n is an odd integer. Then n 2k + 1 for some integer k. Then %3D 7n - 5 = 7(2k+1) -…

A: Theorem If n be an odd integer the 7n-5 be an even integer.