a. Show that for every non-negative integers m, n, N = 7.5" has exactly (m + 1)(n+1) different positive divisors.

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

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Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: Prove that the product of any k consecutive positive integers is an integer multiple of k!.

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Q: Show that for positive integer n,

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

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

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Q: n(nº +5) is divisible by 6, for each integer n 0.

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Q: 9. Prove that n - n is divisible by 30 for every integer n. te

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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: exp(ix())= 4k+1 2

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Q: Prove that for any integer n, we have n - 2 is not divisible by 4.

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Q: 5. Show that 2 32 3" 31 3" for all positive integers n.

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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>1. All prime divisors of n!+1 must be bigger than n. This is…

Q: Show that for every positive

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

A: Introduction: Consider a starting point for which the assertion is correct. The proposition is true…

Q: 8) Prove that n³ + 2n is divisible by 3 for any natural n. 5

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Q: B. Show that 4n+ 4n+1 + 4n+2 + 4n+3 + 4n+4 is divisible by 11 for any positive integer n. C. Find…

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Q: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

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

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Q: For every integer n > 0, 1+2+22 + · + 2" = 2"+1 – 1.

A: Given: The statement 1+2+22+⋯+2n=2n+1-1 Proof: Base step: Show the statement is true when n=0.…

Q: For all integers n> 0, 32n – 1 is divisible by 8.

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Q: 7. Show that for every natural number n2 1, the integer ) + () +*(;) + - +) +22 +... +2" is…

A: Using binomial theorem:  n0+2n1+22n2....2nnn=n01n·20+n11n-1·21+n21n-2·22....nn1n-n·2n…

Q: 10n+1 + 10" +1

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Q: Prove that there is no integer n for such that 5n+3 is divisible by 5.

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Q: prove that it ņ is an 3n +2 is odd then integer and is add

A: This is related to Number System

Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: 3. Prove that when you select n+1 numbers from {1,2,...2n} two of them Must be consecutive (that is,…

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Q: 10. Prove that n³-n is divisible by 6 for every integer n.

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Q: Prove that 9? −1 is divisible by 8,for all n ≥ 1

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Q: Show that if 2n−1 is prime then n is prime.

A: We have to show that if 2n-1 is prime then n is prime.

Q: Prove that n2 + 4 is not divisible by 3 for all integers n.

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Q: Find all positive integers n such that the sum of all positive divisors of n is odd.

A: Given the data