Question 3 (a) Write down the prime factorization of 10!. (b) Find the number of positive...
8. (a) Prove that if p and q are prime numbers then p2 + pq is not a perfect square. (b) Prove that, for every integer a and every prime p, if p | a then ged(a,pb) = god(a,b). Is the converse of this statement true? Explain why or why not. (c) Prove that, for every non-zero integer n, the sum of all (positive or negative) divisors of n is equal to zero. 9. Let a and b be integers...
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...
1. (10 points) For the following questions, let p, q, r e Z be distinct positive prime integers, and define n=p?q?r. (a) How many distinct positive divisors does n = pq?r have? When counting positive divisors, do not count 1, but do count n itself (b) Using a result in the book, justify that n does not have any additional divisors beyond those given in (a).
8. Define (n) to be the number of positive integers less than n and n. That is, (n) = {x e Z; 1 < x< n and gcd(x, n) = 1}|. Notice that U (n) |= ¢(n). For example U( 10) = {1, 3,7, 9} and therefore (10)= 4. It is well known that (n) is multiplicative. That is, if m, n are (mn) (m)¢(n). In general, (p") p" -p Also it's well known that there are relatively prime, then...
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.
(For this question, do not use prime factorization) Suppose that a, b and d are positive integers with d ab. Prove that there exists positive integers e and f such that ea, f b and d= ef. Further show that the values of e and f are unique if (a, b) = 1.
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...
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.
7.23 Theorem. Let p be a prime congruent to 3 modulo 4. Let a be a natural number with 1 a< p-1. Then a is a quadrutic residue modulo pif and only ifp-a is a quadratic non-residue modulo p. 7.24 Theorem. Let p be a prime of the form p odd prime. Then p 3 (mod 4). 241 where q is an The next theorem describes the symmetry between primitive roots and quadratic residues for primes arising from odd Sophie...
This Question must be proven using mathematical induction 1: procedure GCD(a, b: positive integers) 2 if a b then return a 3: 4: else if a b then 5: return GCD (a -b, b) 6: else return GCD(a,b-a) 8: end procedure Let P(a, b) be the statement: GCD(a, b)-ged(a,b). Prove that P(a, b) is true for all positive integer a and b.