7) Prove that for any integer n, n' -n is even.

Q: let 2^k+1 be a prime number. prove that then k=0 or k=2^n for some n =0

A: We have given that , 2k+1  be a prime number . We need to prove that , k = 0  or  k = 2n When k = 0…

Q: Let denote the n-th prime number. Show that Pn Pn n=1 diverges.

A: Divergence of the series

Q: Prove that the analogue of lemma 1.23 is not true for numbers of the form 4n+3 where n is an…

A:

Q: Show that there are infinitely many primes of the form 4k + 3.

A: We have to show that there are  infinitely many primes of the form 4k+3.

Q: This exercise uses one or more of the following results from Section 5.2. Exercise 5.2.10: For every…

A: We will use the given information to complete the incomplete proof of given statement.

Q: What is the smallest integer that is a quadratic residue for all prime moduli p > 2?

A: We know if there is an integer 0<x<p, where p is a prime, such that x2≡qmod p, then q is said…

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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Q: 10. Show that the interval (0, 1) is covered by the open intervals In (1/n, 1- 1/n), n 2 3, but (0,…

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Q: Prove that 4 < n! for all n 2 10 by induction

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Q: What is the smallest positive integer k such that xk = e for all x inU(7 . 17)? Generalize to U(pq)…

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Q: 1 Prove that + n is an integer for any n E N. 6

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Q: 1. Show that if m and n are odd integers, then m+n is an even intege

A:

Q: Prove n (2k – 1) = n² - k=1 for every n EN.

A:

Q: Suppose n is an integer and consider the number n* – 6n³ – 18n² + 6n + 1. a) By expanding the…

A: Explanation: a. Given that,                    Expand the right side and show that both sides are…

Q: Suppose n ≥ 2 is an integer with the property that whenever a prime p divides n, p 2 also divides n…

A: let n=p1a1p2a2....pmam be the prime factorization of n  Here p1,p2....pm are all distinct primes and…

Q: 3. Prove that if n is an odd integer then 7n+5 is even.

A:

Q: Suppose that n is an integer such that n≥3. Using a combinatorial proof, show that:

A:

Q: 4. Suppose we are in a parallel universe where 6 dollar coins and 11 dollar coins exist. Show that…

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Q: Suppose n₁, n₂ are positive integers and gcd(01, n₂) = 1. Then for every integer x, if n₁x and n₂|x,…

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Q: Prove that n!+1 and (n+1)!+1 integer n. relatively prime for every positive

A: Given, n!+1 and n+1!+1are two numbers and to prove that they are relatively prime.

Q: Prove bey wnduc hiou that for n22.

A:

Q: 1. Show for all natural numbers n that (2k – 1) = 1+3+ ...+ (2n – 1) = n². k=1

A:

Q: Prove the following statement directly from the definitions. The product of any two even integers is…

A: The product of any two even integers is a multiple of 4.

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

A: This is related to Number System

Q: 6. Verify Euler's Theorem for n = 16 and a = 5. (7 from 6.5)

A:

Q: (3n)! is an integer for all n > 0. (3!)n Prove that

A: We will prove it by Mathematical induction. First We show it is true for initial condition n=0…

Q: Prove by induction  that the sum of the first n even integers starting at 2 is n(n+1).  (That is,…

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Q: 4. Let n and k be two positive integers with n 2. Prove that (n 1)² divides n-1 if and only if n -1…

A: The given problem is to prove the given following statement. Given statement, let n and k be two…

Q: Suppose that m, n are two integers such that m3 – 2n2 = 1. Does it follow that m and n are…

A:

Q: Prove by cases that for all integers n, n +5n is even. Case 1 (n is even): Case 2 (n is odd)…

A: The given term is n2+5n

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: Let n and k be integers with 1<k<n. Show that n n+1 Zk=1 (k - - n

A: Given Solution:

Q: Prove that 11 is the only prime number of the form 4n2 - 25.

A:

Q: If nis a positive integer 1 then prove (D - a) n!

A:

Q: first prove that ()" <n! < ()" forn2 6.

A: We will prove                        n3n≤n!≤n3n for n≥6.

Q: 5. Prove that for all n EN sufficiently large, 3" > 2n² + 1.

A:

Q: (i) Prove that there are infinitely many primes of the form p = 6n – 1. -

A: Given:    Infinitely many primes of the form p=6n-1     Proof:         Suppose p1, . . . , pn are…

Q: 2. Prove that if 2n² + 7n is even and n is an integer, then n is even.

A: We have to prove that the given statement is true by the help of mathematical induction

Q: Prove that by mathematically Induction (Marks = 05) 2n < (n + 1)!, for all integers n ≥ 2.

A: Answer

Q: 1. Show that 2n n=0 onverges absolutely.

A: Given series: According to the ratio test:

Q: Find all positive integers n such that both n + 2020 and n + 2021 integers. are LVn +1]

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Q: Prove that n^2 is an even number if and only if 3n + 1 is an odd number by proving  If n^2 is even…

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Q: 4. Prove that 3"< n! if n is an integer greater than 6.

A: As per our guideline we are supposed to answer only one asked question.kindly repost other question.…

Q: prove that n^2 +1 >= 2^n when is a positive integer with 1<=n<=4

A: To prove n^2 +1 &gt;= 2^n when is a positive integer with 1&lt;=n&lt;=4

Q: Prove by direct method: for any n integer,if n is odd then 3n+2 is odd

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Q: Suppose that m,n are two integers such that m3−2n2=1. Does it follow that m and n are relatively…

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Q: Show that the sum of the first n positive odd integers, 1 + 3 + 5 + .......+ (2n - 1), is n2.

A: Given: 1+3+5+...........+2n-1 is n2

Q: Prove that (1 -)(1-÷)(1-→).--(-) for all integers m 2 2 by m 2m nathematical induction.

A: *concept:- formula, a^2 - b^2 = (a+b)*(a-b).

Q: 3. Prove that 0 divides an integer a if and only if a 0.(In your proof. you can freely ILNE your…

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.