1. Prove that, for any positive integer n," n3 +2n is divisible by 3 " using the method of mathematical induction.2. Prove that " The sum of two odd numbers is even: using direct proof.3. Let n be an integer, show that if n? is even, then n is also even using indirect proof.4. Check whether the following proposition is a tautology, contradiction, or contingency.Ip vg)n (p +r)^ (g → r)] →r5. Prove by the principle of mathematical induction. show that 1+2+22+...+ 2" = 2" +1-1 for all non negative integers n.

Multiple: 1) We need to prove that , for any positive integer n , n3 + 2n is divisible by 3. We…

Answered: 1. Prove that, for any positive integer…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1. Induction and prime factorisation of natural numbers: Let n E N be a natural number. A prime factorisation of n is an equation n = prime number and d; e {0}UN. Using Mathematical Induction, prove that every natural number has a prime factorisation. di d2 P1 P2 de ..Pk where for i = 1, . . . , k, The following statements inay be useful in your proof: (Q1) Vm, n E N (m 2) →
Prove the following proposition using forward-backward method, "If n is an even integer, then (n² + 2) is divisible by 2". Provide both the analysis of proof and the condensed proof.
- A Pythagorean prime number is a prime number of the form 4n + 1, where n ≥ 1. A Mersenne prime number is a prime number of the form 2" – 1. A perfect number is a number that is equal to the sum of all of its divisors, including 1 but excluding the number itself. Consider the following statements: (1) Let m is the smallest Pythagorean prime number. Then, 2¹ – 1 is a Mersenne prime number. (2) If m is a perfect number, then 2 – 1 is a Mersenne prime number. Select one of the following choices: (a) (1) is True and (2) is False. (c) (1) and (2) are True. (b) (1) is False and (2) is True. (d) (1) and (2) are False.
(a) Consider the following proposition concerning an integer n > 2. If 3" – 1 is divisible by 4 then n is an even number. (i) Write down the contrapositive of this statement. (ii) Is the proposition true? Justify your answer!
12. Consider the following (true) claim and two possible proofs. Claim. If n is a positive integer, then n² + 3n+2 is not a prime number. Proof 1. Note that n² + 3n+2= (n + 2)(n+1). Since n 1, we have n + 1>1 and n+2 > 1, so n²+3n+2 has at least two divisors other than 1 and is therefore not prime. Proof 2. If n is even then we can write n = 2k for some positive integer k. Then n² + 3n+2 = 4k² + 6k+2 = 2(2k² +3k+1), which is even and greater than 2. Since 2 is the only even prime, it follows that n² + 3n+ 2 is not prime. Which of these proofs are valid (i.e. which of them actually prove the claim)? (a) Both proofs are valid. (b) Proof 1 is valid, but Proof 2 is not. (c) Proof 2 is valid, but Proof 1 is not. (d) Neither proof is valid.
14) (Optional and very challenging.) Prove that for any nonnegative integer n, if the sum of the digits in n is divisible by 3, then n is divisible by 3. Hint: Let n be arɑk–1 •*· ɑo, where aq, a1, .….. , ar are the respective digits of n and let s denote the sum of the digits of n. Consider n – s.
what steps need more justification or explanation Proof: 1) Proof by contradiction, so I need to show that the negation is false. 2) Negation: There exists a largest prime number N. 3) Suppose there exists a largest prime number N. Then all numbers bigger than N can be factored into some smaller primes. 4) Consider the product N!+1= (1*2*3*4 … *N) +1 5) Any number between 1 and N will have a remainder of 1 when you try to divide it into N! +1. 6) Rephrasing: (N!+1)/x is not a whole number for any integer x in {1, … N}. 7) That means none of those numbers {1, … ,N} can be factors of N!+1 because they don’t divide in evenly. 8) So N!+1 has no factors between 1 and N. But all of the primes are between 1 and N because N is the largest prime. So N!+1 would need to be a prime! 9) But N!+1 is bigger than N. Contradiction
Consider the following statement. For every integer m, 7m + 4 is not divisible by 7. Construct a proof for the statement by selecting sentences from the following scrambled list and putting them in the correct order.

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