Which of the following correctly describes the Fundamental Theorem of Arithmetic? Every positive integer greater than 1 is either a prime or a product of primes, unique up to rearrangement of factors. There are infinitely many primes. Every positive integer greater than 1 is either a prime or divisible by a prime. Every positive integer greater than 1 has a prime divisor.

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 the theorems: i) "If n is an integer, and if n² is even then n is even". ii) "If n is an…

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Q: Every integer n22 either is a prime or can be expressed of a product of primes. This statement is…

A: Every integer n≥2 is a prime or can be expressed as product of primes. This statement is Fundamental…

Q: Prove that for a positive integer n we have that  33n+3 - 26n - 27 is divisible by 169.

A: To show  33n+3 - 26n - 27 is divisible by 169.

Q: Prove that for every non-negative integer n the value of n² + n+41 is prime ?

A: The number n² + n + 41 is not prime for every natural number n. Let, f(n) = n² + n + 41

Q: A. Use Mathematical Induction to prove that: (n+1)(2n+1)(2n+3) 12 + 32 + 52 +...+(2n + 1)? 3…

A: The following two steps are used to prove a statement P(n) by the principle of mathematical…

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

A: 11·2+12·3+13·4+---+1nn+1=nn+1 Let us prove this statement by using the Principle of Mathematical…

Q: Let n be an integer. Prove that if 2n +4 is divisible by 4, then n is even.

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Q: 2. Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? +1 is divisible by 10. 3.…

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Q: Let a and b be two integers such that b | a. Let p be a prime such that p|a and p{b. Prove that p|…

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Q: 4. Prove that every odd natural number is of one of the forms 4n+ 1 or 4n+3, where n is an integer.…

A: Disclaimer:  Since you have asked multiple question, we will solve the first question (Number 4) for…

Q: (a) Use mathematical Induction to prove that 4n + 7" + 10 is divisible by 3 for every non-negative…

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

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

A: Given arithmetic progression (AP) is: 2+5+8+.........+3n-1 Here the first term is 2 and the common…

Q: 24. Determine whether each of these integers is prime, verifying some of Mersenne's claims a) 2 -1…

A: As per guidelines, we will solve first three subparts of the question a) Given integer is 27-1  We…

Q: 2. Give an example to show that the following conjecture is not true: Every positive integer can be…

A: Every positive integer can be written in the form p+a2, where p is either a prime or 1, and a≥0 We…

Q: 6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Use mathematical induction to prove that each of the following is true for all natural numbers n. 1.…

A: For n=1, 2=1(1+1)2=2 This means n=1 satisfies the given statement.    Let us assume that the…

Q: Use the following statement to answer the following questions. Prove that 11"-6 is divisible by 5…

A: See the explanation below and follow the same if you understand the explanation please give ratings.

Q: Prove that if  n  is a positive integer, then  5 | (6^n − 1)

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Q: Label each of the following statements as either true or false. The set of prime numbers is closed…

A: Consider a set of prime numbers    A set is closed with respect to multiplication if for all a,b…

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: Show that for every positive

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Q: 3. Let a₁ = 2, a2 = 1, and for each natural number n > 2, let an 2an-1-an-2. Prove that for all…

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Q: 5. Every positive integer is the sum of distinct powers of three.

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Q: How to solve this?

A: According to the given information, let p be a prime number such that p>2. It is required to…

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

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Q: 2. Using the Fundamental Theorem of Arithmetic, prove that for all a, b, c E Z, if alc, blc and (a,…

A: Every integer can be factored uniquely into primes upto order of arrangement. (a,b)=1. Then there…

Q: Prove that there is a natural number N such that for all natural numbers n > N, 2/n < 0.1.

A: Assume that there does not exist N such that for any natural number n,  n&gt;N, 2/n&lt;0.1.

Q: Prove that “if n3 + 5 is odd, then n is even” by using the following: a) Direct Proof b) Indirect…

A: If n3 +5 is odd then n is even. Given:  n3 +5 = 2K+1                (K is any integer)       n3  =…

Q: Prove that every prime greater than 3 can be written in the form 6n + 1 or 6n + 5.

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Q: Let p be a prime number such that p> 2. Use the proof by contradiction to show that at least one of…

A: See the solution attentively I think you will understand easily.

Q: Prove by Mathematical Induction that for any natural number n the number 52n+1 – 22n+1 is divisible…

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Q: Let n be any integer. Prove that 3n? – 2 is odd if and only if n is odd.

A: Answer

Q: . Let n be an integer. Prove that n is odd if and only if 3n? – 7 is even.

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Q: A sequence {am} of real numbers is defined recursively by a1 = 1, a2 = 2 and a, = (i–1)a; for n > 3.…

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Q: show that no integer of the form n^3+1 is a prime other than 2

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Q: Prove or disprove that for every positive integer n, if n is even then n^{2}+1 is prime.

A: False

Q: Prove or disprove: If three consecutive integers are multiplied together, and the second in order of…

A: We first look out some examples to understand the fact- choose three consecutive numbers be 1,2,3.…

Q: Prove, using any method, the following statement: if n is an integer, then (3n+1)(5n+4) is an even…

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Q: Let n ∈ N and p be a prime. Prove that if n is not divisible by p then n and p arerelatively prime

A: From Prime factorization theorem we know that any natural number can be written as product of…

Q: 11. Prove that for each integer a, if a? - 1 is even, then 4 divides a?

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Q: 3. Write out in set notation: a. The set of all integers divisible by 3 b. The set of all primes…

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Q: Which of the following is the best way to PROVE or DISPROVE the statement "Every integer greater…

A: We have to check

Q: Let n > 1 be an integer. Show that if pn for all primes p≤ √n, then n is either prime or the product…

A: Introduction: Proof by contradiction, sometimes referred to as indirect proving or the reductio ad…

Q: Prove that every positive integer greater than 1 has a prime divisor. b.) Prove that there are no

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Q: Let n be an integer. Show that if n 2 is a multiple of 3, then n must also be a multiple of 3. (a)…

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Q: 5. (a) Use modular arithmetic to show that if an integer a is not divisible by 3, then a2 = 1 (mod…

A: To prove the given statements (using arithmetic modulo 3)

Q: Prove that for any integer n, if n³ is odd then n is odd.

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Q: Let n be any integer. Using a proof by cases, prove that n4 is either in the form of 5k or 5k+1 for…

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Q: In an inductive proof that for every positive integer n, n п(п + 1)(2n + 1) Σ 6. j=1 what must be…

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