Let n be a positive integer. Show that nº + 4n +5 has no prime divisor...
Part 15A and 15B (15) Let n E Z+,and let d be a positive divisor of n. Theorem 23.7 tells us that Zn contains exactly one subgroup of order d, but not how many elements Z has of order d. We will determine that number in this exercise. (a) Determine the number of elements in Z12 of each order d. Fill in the table below to compare your answers to the number of integers between 1 and d that are...
Find the smallest positive integer that has precisely n distinct prime divisors. 'Distinct prime divisor'Example: the prime factorization of 8 is 2 * 2 * 2, so it has one distinct prime divisor. Another: the prime factorization of 12 is 2 * 2 * 3, so it has two distinct prime divisors. A third: 30 = 2 * 3 * 5, which gives it three distinct prime divisors. (n = 24 ⇒ 23768741896345550770650537601358310. From this you conclude that you cannot...
Two positive integers are amicable if each prime divisor of one is a divisor of the other. Example: 6 and 12 are amicable, since each prime divisor of 6 (2 and 3) also divides 12, and each prime divisor of 12 (again 2 and 3) divides 6. Another: 12 and 15 are not amicable, since a prime divisor of 15 (namely 5) does not divide 12. Find the sum of all amicable pairs whose two members are both less than...
The prime factorization of a positive integer n is p^3. Which of the following is true? Explain and show your answers. I. n cannot be even II. n has only one positive prime factor. III, n has exactly three distinct factors.
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
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
5. (a) Show that 26 = 1 mod 9. (b) Let m be a positive integer, and let m = 6q+r where q and r are integers with 0 <r < 6. Use (a) and rules of exponents to show that 2" = 2 mod 9 (c) Use (b) to find an s in {0,1,...,8} with 21024 = s mod 9.
please show converse as well 20, Let p be prime. Show that p X n, where n is a positive integer, if and only if ф (np)s (p-1)ф (n).
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 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?