Part III: Prove or Disprove the following 9. For every positive integer n, n2 – 2 is not prime is not necessary for n to be odd.

Q: An integer n is said to be abundant if o(n)>2n. Every multiple of abundant integer is abundant *…

A: We know,  abundant number is a number that is smaller than the sum of its proper divisors. An…

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

A: Given, 3+9+15+..+(6n-3)=3n2

Q: If n is a positive integer with (n + 1)(n + 3) is odd, then (n + 2)(n+ 4) must be a multiple of 1) 6…

A: Given:  (n+1)(n+3) is odd We will get an odd number from product of two numbers only if the two…

Q: If n is an even postive integer, then 3n+2^n-2 is odd.

A: Number problem....

Q: Prove the the following using either direct proof, indirect proof, or proof by contradiction. 2. n…

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Q: 4. Use the PMI to prove the following for all natural numbers n. |(c) E-1 2' 2n+1 – 2 i=D1

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Q: 1. If n is an integer, then 2n +1 is odd.

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Q: Question 2 Prove the statement:: "For all integers n, if 3n+ 5 is odd, then n is even." (Hint: You…

A: The contrapositive of a statement is a proposition formed by contradicting both the  hypothesis and…

Q: 1. Give the direct proof of the following. a. If n is an odd integer, then n2 is odd.

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Q: Prove that the statement is true for every positive integer n.n < 2n

A: n<2n Let us prove this statement by using the Principle of Mathematical Induction. When n=1,…

Q: 1. Let n be an integer. Prove by contradiction that if n? + 5 is odd, then n is even.

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Q: 'Disprove the statement: For any n E N, if n² +3 is prime, then 2n+7 is prime.

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Q: If n is an integer, then 2n + 1 is odd

A: Given , n is an integer We have to show that 2n+ 1 is odd

Q: Prove that if n is an integer, then 3n? + n + 14 is even

A: Case I - Let n be an odd integer , then n= 2k+1 where k is an integer. Then 3n2+n+14   =…

Q: The small positive integer n for which (1+i)?"=(1-i)?n is =..

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Q: Prove or disprove: for every natural number n,2+5+8+...+(3n-1)=n(3n+1)/2

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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: Is the following statement true or false? For every integer n, if n is prime then (-1)^= -1. Which…

A: The given statement is: For every integer n, if n is prime then -1n=-1. We know that a non negative…

Q: Disprove the claim: 3 m E N such that m > 3 and m? – 1 is a prime number.

A: The claim is given : ∃ m∈N such that m≥3 and m2-1 is a prime number Now, m2-1=(m-1)(m+1)--(1) Case…

Q: 1. Prove the following statements below. Consider n to be an integer. (a) If 3n + 2 is an odd…

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Q: Prove that the statements here are true for every positive integer n.   1 · 2 + 2 · 3 + 3 · 4 + · ·…

A: To prove: For every positive integer n, 1.2+2.3+3.4+.......+n(n+1)=n(n+1)(n+2)3

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: Prove or disprove: 3n = ω(2^n ).

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Q: If N is an odd integer, then (-1)^N = -1

A: Suppose n is an integer. If n is even, then n=2m for some integer m. If n is odd, then n =2m+1 for…

Q: The number of primes of the form n² – 6n + 5 where n is an integer is O 2 O 3 O 1

A: We have n2 -6n+5 ⇒n2-5n-n+5⇒n(n-5)-(n-5)⇒(n-1)(n-5)

Q: Do all four problems. 1. Prove that there exists a unique prime number of the form n2 – 4, where n…

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Q: 1. Prove the following statements below. Consider n to be an integer. (a) If 3n + 2 is an odd…

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Q: Show that if n is a positive integer, then (n + 1, n2 - n + 1) = 1 or 3

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Q: 2. Prove that if n is even, then 3n + 1 is odd

A: The given expression is 3n + 1. Prove that 3n + 1 is odd, if n is even.

Q: Show that if n is a positive integer, then k+4 n(3n +7) k(k +1)(k+2)¯ 2(n+1)(n+2) k=1

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Q: For every integer n23 I+ 2+ 2?+ +4^= 4(4n-16) %3D 3. Proof (by induction)

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Q: 1. Let n be an integer. Prove that if n is even, then n2 + 2n + 3 is odd.

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Q: 15. Prove that for each natural number n, s+ (4n-3)(4n+1) 4n+1 ...

A: Given The given expression is 11·5+15·9+·  ·  ·+14n-34n+1=n4n+1

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

A: Prove that the statement is true for every positive integer n. 22+42+62+...+2n2=2n2n+1n+13 We prove…

Q: If n is a perfect number, prove Σ d|n 1 2.

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Q: Disprove the claim: 3 m E N such that m 2 3 and m? – 1 is a prime number. -

A: Given that ∃m∈ℕ such that m≥3 and m2-1 is a prime number. Take m=4. It is known that 4∈N And 4>3.

Q: If an integer n is odd, then n2 + 2n +1 is even. Select one: True O False

A: Given n is odd Now n2 + 2n + 1 = (n+1)2

Q: 1. Prove that if 7n2 + 5 is even, then n is odd

A: The given statement is,

Q: 11. If n is an odd integer, show that n + 4n² + 11 is of the form 16k.

A: If n is an an odd integer then we have to show that n4 + 4n2 + 11 is of the form 16k for some…

Q: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even.

A: Given: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even

Q: Answer the following questions: (1)Prove by using Contraposition to show that: If (n + 1)² is odd…

A: Part 1 To prove a statement "if p, then q" by contraposition, we prove "if not q, then not p" Now,…

Q: Prove that the statements here are true for every positive integer n. 12 + 22 + 32 + · · · + n2=n(n…

A: Proved that for n≥1 fn=12+22+32+⋯⋯+n2=nn+12n+16 For n=1 LHS=12=1 RHS=nn+12n+16=11+12+16=66=1…

Q: 13. Prove or disprove the following by mathematical induction, where n is set of positive integers.…

A: ..

Q: Prove the following:For every natural number n≥1, 1+4+7+⋯+(3n−2)=n(3n−1)2.

A: To prove the identity involving the sum to n terms of the given series

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

A: 13+23+33+---+n3=nn+122 Let us use the Principle of Mathematical Induction to prove the given…

Q: (4a) Let n E Z. Prove that if n is even, then -5n – 3 is odd.

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Q: give direct proof by contradiction that is 5n-1 is odd, then n is even

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Q: Prove the following:For every natural number n≥1, 1+4+7+⋯+(3n−2)=n(3n−1)2.

A: To prove the identity giving the sum of the series