How many integers k with 1 S k < 550 have GCD(k, 550)- 1? If n 〉 2 is an integer with o(n) -1, e...
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 3 Let n and k > l be positive integers. How many different integer solutions are there to x1 +...+ In = k, with all xi <l?
Consider the function 0 : 2+ + 2+ $(n) = number of integers k, 1 <k <n that are relatively prime with n (that is such that (k, n)-1) If n is a prime number º(n) = n n-1 O 1
Given the values of n below, determine, without exhaustive search, etc., how many integers k there are, with gcd(k, n) = 1, and 1 <= k <= n, such that k has a square root modulo n. Do this for (a) n = 143, (b) n = 286, (c) n = 572, (d) n = 1144, and (e) n = 2288. In each case, determine also phi(n), so as to be able to tell what fraction of reduced residue classes...
Exercise 6: How many positive integers between 1 and 10", with n 1 a positive integer, does not have 7 as a digit in their base 10 representation?
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
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1...
(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...
2. (15 points) Prove that for a positive integer n, the number gcd (n + 1, na — n + 1) is equal either to 1 or to 3.