230hw4 - ,p 1,p 2,p k Then consider the number N = 4 p 1 p...

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Math 230 E Fall 2011 Homework 4 Drew Armstrong Problem 1. (a) Consider a,b,d Z with gcd( a,d ) = 1. Prove: If d | ab then d | b . (b) Consider a,b,m Z with gcd( a,b ) = 1. Prove: If a | m and b | m then ab | m . (c) Consider a,b,c,n Z with gcd( c,n ) = 1. Prove: If ca cb mod n then a b mod n . (Hint: Part (a) is just like Euclid’s Lemma (Theorem 2.53). Then you can quote part (a) to prove (b) and (c).) Problem 2. (a) Consider an integer n > 1. Prove that if n 3 mod 4 then n has a prime factor of the form p 3 mod 4. (Hint: You may assume Proposition 2.51.) (b) Prove that there are infinitely many prime numbers of the form p 3 mod 4. (Hint: Suppose there are only finitely many and call them 3
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Unformatted text preview: ,p 1 ,p 2 ,...,p k . Then consider the number N = 4 p 1 p 2 ··· p k + 3. Apply part (a).) Problem 3. Let d be a common divisor of a,b,n ∈ Z . Prove that we have ax ≡ b mod n ⇐⇒ a d x ≡ b d mod n d . Problem 4. Solve the congruence 1713 x ≡ 871 mod 2000 for x . (Hint: Apply the Extended Euclidean Algorithm 2.25 to the numbers 1713 and 2000.) Problem 5. Given integers a,n ∈ Z we define the set [ a ] n := { a + nk : k ∈ Z } ⊆ Z . For all a,b ∈ Z , prove that [ a ] n = [ b ] n ⇐⇒ a ≡ b mod n....
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This note was uploaded on 01/08/2012 for the course MATH 461 taught by Professor Armstrong during the Fall '11 term at University of Miami.

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