Prove the following mathematical statements: Write proofs for the given statements, (inserting parenthetic remarks to explain the rationale behind each step)  a) The product of an even number and any other number is even. (use direct proof)  b) If n>0 and 4n−1 is prime, then n is odd. (use contradiction or indirect proof)  c) The sum of the first n numbers is n(n+1)/2. (use mathematical induction)

Q: Letting aa and bb be two positive real numbes such that ab=lab=1, which of the followings is true?…

A:

Q: (1) What is gcd(31212021 + 3, 31212021 - 3)? I am aware of 3121 is a prime number, and 2021 is…

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: (a). From the set of numbers { -3, 0, √19, √-9, 1/3, 0.6, 7i }; identify the number/s which is/are:…

A:

Q: Prove the statement. "For any integer n, nº +4 is not divisible by 8." by Contradiction.

A: Given: n be any integer. To prove: n3+4 is not divisible by 8.

Q: Use proof by contradiction to show that, "The sum(s) of an irrational number(i) and a rational…

A:

Q: Claim: For every odd integer n, n² - 1 is a multiple of 8. Purported proof: Suppose there is an…

A: The given claim is "For every odd integer n, n2−1 is a multiple of 8".Purported proof:Suppose there…

Q: Give an example of (a) a composite number of the form 2º – 1, where p is an odd prime. (b) a…

A: b) Here we need to find an example of a composite number of the form  22n+1 where n is a non…

Q: Tell whether each number is an element of  (a natural number),  (an integer),  (a rational number),…

A: Since we only answer up to 3 sub-parts, we'll answer the first 3. Please resubmit the question and…

Q: 7. Use the properties of the operations of rational number arithmetic to perform these calculations.…

A:

Q: Prove the following statement directly from the definition of rational number. The difference of any…

A: We need to prove that the difference of any two rational numbers is a rational numbers.We know that…

Q: Consider the two statements: P: For every integer n there exists an integer m such that m + n = 0.…

A:

Q: If m and n are rational numbers such that √m +√n = √5+√24 ,calcule a possible value for m² + n²

A: Given that √m +√n = √5+√24 To calcule a possible value for m² + n²

Q: Number 4 only. Thanks!

A:

Q: How do I prove: If n > 1 is a natural number then there is a prime number p such that p|n.

A: This is a theorem of real analysis in which we have to find that there is a p who divides n.

Q: Find the sum 1 9k2 + 12k – 5 k=1 Express your answer as a rational expression in terms of n.

A:

Q: * 4. Determine the conditions under which each of the following conditional sen- tences will be a…

A: (a) The conditional statement "If a + 2 = 5, then 8 < 5" will be true when the expression "a + 2"…

Q: For each natural aumber a, the sum 1(2) + 2(3) ++ R(n + 1) is equal ta None of the choices This…

A:

Q: Use mathematical induction to prove that the given statement is true. n< 2"

A:

Q: Question: Evaluate each following expressions a.) If x =0   evaluate 4x**2 + 5(9 +x) + x **5 b.)…

A:

Q: Briefly explain. what is the error in the proof of the claim: for every natural number n, n+2=n+4?…

A: Concept Used:(Principle of mathematical Induction) If Pn be the statement n is natural number to…

Q: umbers and q (> 0) the first of n harmonic means between the me numbers. Then prove that

A:

Q: Question:  a) prove that 100 can not be written as sum of an odd integer and two even integers.…

A: a) To prove that 100 cannot be written as sum of an odd integer and two even integer. Even integer…

Q: For the following statements identify the mistakes in the beginning of the proof. Prove: For all…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Find the indicated real nth root(s) of a. n = 4, a = -81 O +3 -/3

A: This question is about nth real roots of numbers

Q: Given the following statement (M): “If b is an irrational number then r = is irrational." None of…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: (а) — 122 (b) - 10 (c) 0 (d) 14 18 (e) /25 (f) /31 (g) 81 From the list above, determine the…

A:

Q: Calculate a) 1 2/7 x (-2  2/3) ÷ 13/7 + 11/3  b)In a GP the 6th term is 972 and the 3rd term is 36.…

A: Part (a) Given expression is 127×-223137+113

Q: There exists a real number R which is greater than 5 O R= {7, 42,2t, and 97/3) O R= {1, 2, and T/6}

A: