Given n > 1, let p be a prime number such that n < p < 2n (by the way,2n( ²1 ) ²?there is always a prime between n and 2n). Does Pndivide

here we need to expand by using nr=n!r!×n-r! and then show that expression obtained have a factor…

Answered: Given n > 1, let p be a prime number…

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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2. Let p ∈ Z be a prime integer.(a) Show that if p > 3, then p is of the form 6k + 1 or 6k −1 for some k ∈ Z.(b) If p > 5, show that dividing p by 10 can only leave remainders of 1, 3, 7, or 9, and findexamples of primes with these remainders.
Do n.9
Let pn be nth prime. prove that the sum 1/p1+1/p2+…..+1/pn is never an integer.
(24) Let p, q E N be two distinct primes. (a) Find the following quantities (possibly in terms of p and q): gcd(p, q), gcd(p²q, pq²), (b) What are all the possible values of gcd(6p, 9q)? Explain. ged(pq, p²q² + 1).
Let n = 2 x3 x 5 ×7 × 11 × 13 + 1. a) What is n ÷ 2, n ÷ 3, n ÷ 5, n ÷ 7, n ÷ 11, and n+ 13? b) What do you notice about the remainders of each of these answers in part a? {pi, P2, .... Pk } be a set of k different prime numbers, and now let n = pi X \p2 x ... \p + 1.Isthereap_i \in Psuchthatp_i \mid n$? Explain. c) Let P :=
Let p and q be prime numbers.(1) If p is not equal to q, how many generators does Zpq have?(2) If p = q, how many generators does Zp2 have
Find all positive integers n that is a product of two distinct primes such that o(n) = 60.
(1) Let p be a prime number. Show that for any integer a the greatest common divisor of p and a is either p (if pla) or is 1 (if p does not divide a).
Use the PMI to prove that 3 - 1 is even for all n E N. Proof. Base case: Since 3¹ - 1 = 2, which is even. Thus the statement is true for n = [Select] Inductive step: Assume that there is a natural number n such that 3 - 1 is even. Then 3-1 = [Select] for some integer k. Then 3" [Select] ✓. On both sides, first multiply by 3, then subtract 1, and simplify to get 3+1 -1 = 2( [Select] which is an [Select] integer. Hence, by the PMI, 3 - 1 is even for all n E N. )
Consider the list of numbers: 2n-1, where n first equals 2, then 3, 4, 5,6,…What is the smallest value of n for which 2n -1 is not a prime number. Show all your work.
Pls. answer these two questions by typing and not by handrwriting
Find the smallest positive integer n such that 5(3² +2²) (3¹ +2¹) (3° +2°)... (3²° +2²) > 9 256

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Given n > 1, let p be a prime number such that n < p < 2n (by the way, 2n there is always a prime between n and 2n). Does p divide (²1) ²