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Problem 1. Let a be any positive integer relatively prime to 10 (so gcd(a, 10) 1)....
Any help is much appreciated :) Let p be a prime, and n a positive integer. Prove that NoTE: This appears to be an infinite sum. Eventulo in fact after a point all of the terms are 0
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
Let m be a positive integer and let a and b be integers relatively prime to m with (ord m a , ord m b) )=1. Prove that ord m (ab)= (ord m a) (ord m b) (Hint: Let k=ord m(a),l=ord m(b), and n=ord m(ab). Then 1≡(ab)^kn≡b^kn mod m. What does this imply about l in relation to kn?
number thoery just need 2 answered 2. Let n be a positive integer. Denote the number of positive integers less than n and rela- tively prime to n by p(n). Let a, b be positive integers such that ged(a,n) god(b,n)-1 Consider the set s, = {(a), (ba), (ba), ) (see Prollern 1). Let s-A]. Show that slp(n). 1. Let a, b, c, and n be positive integers such that gcd(a, n) = gcd(b, n) = gcd(c, n) = 1 If...
Let k and a be two positive integers, such that ak-1 = 1(mod k) and gcd(k, a) = 1. Is k prime or composite? If so why and explain all the steps. Thanks
Problem 5. Let a € {11, 111, 1111, ...} be a positive integer with all 1s in its decimal expansion. Prove that a is not a perfect square. (Hint: look at the value of a mod 4).
Problem (5), 10 points Let a0:a1, a2, be a sequence of positive integers for which ao-1, and a2n2an an+ for n 2 0. Prove that an and an+l are relatively prime for every non-negative integer n. 2n+an for n >0 Problem (5), 10 points Let a0:a1, a2, be a sequence of positive integers for which ao-1, and a2n2an an+ for n 2 0. Prove that an and an+l are relatively prime for every non-negative integer n. 2n+an for n >0
(a) Let n be any positive integer. Briefly explain (no formal proofs) why n > 1 ≡ ¬(n = 1). (b) Recall that a positive integer p is prime iff there do not exist a positive integers n and m, both greater than 1, such that p = nm. (I.e., Prime(p) means ¬∃n ∃m (n > 1 ∧ m > 1 ∧ p = nm).) Give a formal proof of the following: for any prime p, any positive integers n...
6. Let n be any positive integer which n = pq for distinct odd primes p. q for each i, jE{p, q} Let a be an integer with gcd(n, a) 1 which a 1 (modj) Determine r such that a(n) (mod n) and prove your answer.
PYTHON! The Sieve of Eratosthenes THANKS FOR HELP! A prime integer is any integer greater than 1 that is evenly divisible only by itself and 1. The Sieve of Eratosthenes is a method of finding prime numbers. It operates as follows: Create a list with all elements initialized to 1 (true). List elements with prime indexes will remain 1. All other elements will eventually be set to zero. Starting with list element 2, every time a list element is found...