3. Show that for all integers n: a) if d is an integer such that d |n + 8 and d | n² + 1, then d | 65 b) if n is a multiple of 65, then n + 8 and n? +1 are coprime.

Q: 28. Prove that for every integer n ≥ 1, 1+3+5++ (2n-1) (2n + 1)(2n + 3) + ··· + (2n + (2n − 1))* 1 3

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Q: Let n denote an integer. Prove that 5|n and 7|n if and only if 35|n.

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Q: If n2 4 is an integer,prove by induction that n<nl.

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Q: 32. Prove that 3 divides n³ +2n whenever n is a positive integer.

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

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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: 3. Show that √2+√3 is irrational. Show that √n-1+√n+1 is irrational for every integer n ≥ 1.

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

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Q: 2. Prove that the product of any two odd integers is an odd integer.

A: Let x and y be two odd numbers. If a and b are odd, then a and b can be written as: x=2m+1 , y=2n+1…

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: An integer is square-free if it is a product of distinct primes. Show that for each integer k≥ 1…

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Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n&gt;1. All prime divisors of n!+1 must be bigger than n. This is…

Q: a) If d is an integer such that d | n + 5 and d | n² + 1, then d | 26. b) If n is a multiple of 26,…

A: a. Given that d is an integer such that d|n+5 and d|n2+1.

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: Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the remainder…

A: we have to Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the…

Q: If the integer K is greater than or equal to 1 and is odd. Prove that the sum of the k powers of…

A: The N must be a perfect square or double square in order for the total of the k powers of positive…

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: ve or disprove: For all positive integers n, if 771 n, then 7(77n) = 47(n)

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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: Use the definitions of even and odd to prove: For all integers n, n2+3n is even.

A: We have to prove, For all integers n, n2+3n is even.   We prove this by using Principal of…

Q: Prove: “The sum of two even integers is even.”

A: To prove: The sum of two even integers is even. n is an even number if it can be expressed as n =…

Q: Prove by contradiction that there are infinitely many non-negative integers.

A: There are infinitely many non negative intergers. To prove this If number of negative is finite,…

Q: Prove that if a and b are both odd integers, then 8 (2a2+36²).

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Q: _4n + 47). Show that there are infinitely many positive integers n that satisfy the division 23|(n²…

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Q: Suppose n is an integer. Prove that if n is even, then n²+3n+5 is odd.

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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: Prove or disprove: For all odd integers a and b, 4 | (3a² + b²).

A: We have to prove or disprove the given statment.

Q: Prove or disprove: If a and b are odd integers, then 4 | (a – b) or 4 | (a+b). -

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Q: Let a, b be positive integers, and let p be a prime integer. Let the notation n|m means m divides n…

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Q: Prove that 74n+3 + 2 is a multiple of 5 for all non-negative integers n.

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Q: n. Prove that 21|(3n7 + 7n³ + 11n) for all integers

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Q: Show that for every integer n ≥ 8, there exist integers a ≥ 0 and b ≥ 0 such that n = 3a + 5b.

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