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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

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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.
Transcribed Image Text: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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