10. Let n be an even perfect number. Prove that n=2*(2k+1 – 1), for some integer k.

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: Suppose that n pigeons are placed into m holes with m ≥ k(n+ 1) for some positive integer k. Suppose…

A: See the attachment

Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: Prove that the hundreds digit of n20 is even.

A: To Prove that the hundreds digit of n20 is even.

Q: 7. Prove that 4 3m² – n² for all odd integers m and n.

A:

Q: 9. Show that there are infinitely many primes of the form 16k + 1. More generally, show that for any…

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: . Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when…

A: Consider an integer x that has a remainder of 2 when it is divided by 5 Thus x=2 mod 5 0r x=5k+2 It…

Q: show that for all N in N * Σ (2/(n(n+2)=3/2 - 2N+3/(N+1)(N+2)

A:

Q: For the first 5 odd primes, show that Wilsn's theorem holds to be true.

A:

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

A:

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

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Q: Prove that n! > 2" for n 2 4.

A:

Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n>1. All prime divisors of n!+1 must be bigger than n. This is…

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

A:

Q: 4. Prove that if n is an integer and n is a multiple of three (that is, n = 3k for some integer k),…

A:

Q: Show that there are infinitely many primes to prove that there are infinitely many primes of the…

A: This question is related to Number Theory. We solve the result by contradiction.

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: Let n be an integer with n ≥ 2. Suppose that for every prime               p ≤√n, p does not divide…

A: given,                                                  p≤n p doesn't divide n. let n is not a prime…

Q: 1. Prove that the Mersenne number M13 is a prime; hence, the integer n = 2' (213 – 1) is perfect.…

A: handwriting solution

Q: A positive integer n is called superperfect if ơ(ơ(n) = 2n Show that 16 is superperfect.

A:

Q: 11. Prove by mathematical induction that 1 Σ k (k – 1) = (n + 1)n(n – 1) | 3 1sksn

A: Now for n = 1, Now assume that the given statement is true for n = k, where k is some positive…

Q: Prove by induction that 1) 2 and 3 can divide n^3-n for all n ∈ N 2) n^2-7n+15 >=0 when n>=3 for…

A:

Q: Show that there are infinitely many primes of the form 6k + 5.

A:

Q: 6. Prove that 2n +5 and 3n+7 are relatively prime for every integer n.

A: What is Relatively Prime Numbers: Relatively prime or coprime numbers are two such numbers which…

Q: 5. Use the direct proof to show that " If n is an odd integer, then n2 + 3 is even

A:

Q: A positive integer n is called superperfect if ơ(ơ(n)) = 2n Show that if n = 2q, where 2q+1 - 1 is…

A: Given: A positive integer n is called superperfect if σσn=2n and n=2q, where 2q+1-1 is prime. To…

Q: Let 2k +1 be a prime number. Prove that then k = 0 or k = 2" for some n > 0. %3D

A: Solution:- Let 2k+1 be a prime number k=0 or k=2n for some n≥0   For, k=0 2k+1=20+1 2k+1=1+1…

Q: 2. What are the two largest positive integers n satisfying n - 1 | n²-A?

A:

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

A:

Q: Show that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer.

A:

Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

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: Use a direct proof to show that the sum of two odd integers(k and l) is even.

A: The number n is even if it is expressed as  n=2m , where m is an integer. The number n is odd if it…

Q: о Prove 2 (2n – 1) = n n - п - 1 k=1

A:

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

A: Given Solution:

Q: 4. Show that P(n,k)= P(n – 1, k)+ k · P(n – 1,k – 1). Using Combinatorial Proof.

A:

Q: 9. Let n be an odd number with exactly four distinct prime factors. If (3, n) = 1, prove that n is…

A: We will use knowledge of number theory to answer this question.

Q: Prove: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

Q: 4. Prove by induction that 21 divides 4"+1 + 52n-1 for all positive integers n.

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

Q: Negate the statement: For all positive integers, n, there exists a positive integer, k, so that…

A: The negation of a statement is:- Statement:- p implies q. Negation:- p does not imply q.

Q: 21. Prove that if n and k are natural numbers such that k < n, then (*:) - (-) - () n k k

A:

Q: 1. How many combinations are there to pick r objects from 2n objects numbered from 1 to 2n with n >…

A: For the solution of the problem follow the next step.

Q: ind the number of ways to choose k children from a group of n children arranged in a circle so that…

A: Consider the given: Let n and k be positive integers with . Choose the number of ways in which k…

Q: Prove that if n > 1 is a number with φ(n) = n/k, then k = 2,3.

A:

Q: 4. For natural numbers n and k, prove (C) - ()

A: We have to prove

Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 < n < 20. Circle any numbers that…

A: Table of values for 10n-1 and 10n+1 for n between 1 and 20. n 10n-1 10n+1 1 9 11…

Q: 5. Find all integers 100 ≤ n ≤ 999 such that the last three digits of n² is same as n.

A: To find: All integers  100≤n≤999 such that the last three digits of n2 is same as n.