Please solve all parts as soon as possible 1. Let a, b eN and n =gcd(a, b). For some prime p, if p divides both a and b, then p divides n.2. Let a, b eN and n = gcd(a, b). Let a = 4 and 3 = . Prove gcd(a, B) = 1.3. Let p and q be primes such that p # q. Prove that for any positive integer n, m, gcd(p", q™) = 1.4. Let n, p e N such that p is prime, p< n and p does not divide n. Prove that the order of p inthe group Zn is n.5. Let m, n e N such that m < n. Prove that if m divides n, then the order of m in the groupZn is strictly less than n.

According to the guidelines I need to answer only 1 question at a time. I still solved first two…

Answered: 1. Let a,b e N and n = gcd(a, b). For…

1. Let a,b e N and n = gcd(a, b). For some prime p, if p divides both a and b, then p divides 2. Let a, b e N and n = ged(a, b). Let a = 4 and 3 = . Prove gcd(a, ß) = 1. %3D

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.8: Introduction To Cryptography (optional)

Problem 23E

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Question

Please solve all parts as soon as possible

1. Let a, b eN and n =
gcd(a, b). For some prime p, if p divides both a and b, then p divides n.
2. Let a, b eN and n = gcd(a, b). Let a = 4 and 3 = . Prove gcd(a, B) = 1.
3. Let p and q be primes such that p # q. Prove that for any positive integer n, m, gcd(p", q™) = 1.
4. Let n, p e N such that p is prime, p< n and p does not divide n. Prove that the order of p in
the group Zn is n.
5. Let m, n e N such that m < n. Prove that if m divides n, then the order of m in the group
Zn is strictly less than n.

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