210hw7 - d ). *Hint for the hint: You know certain things...

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Math 210 Homework 7 Due Monday April 11th 1. Using the suggestions below, construct a proof of the statement: “If a and b are positive integers, then gcd( a,b ) is the smallest positive integer that can be written as xa + yb where x and y are integers.” (i) Show that some positive integers can be written as xa + yb . (ii) By the Well-Ordering Principle, conclude that there is a smallest positive integer that can be written as xa + yb . Give it a name, say d . (iii) Show that d divides a and d divides b . (Hint: “Try” to divide a by d . There will be a quotient and a remainder.*) (iv) Show that if d 2 is any other positive integer that divides both a and b , then d 2 divides d (so d 2
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Unformatted text preview: d ). *Hint for the hint: You know certain things are linear combinations of a and b . Can a small positive number be a linear combination of a and b ? 2. Prove that if p > 3 is prime, then 24 | p 2-1. 3. If n Z and 5-n , show that n 2 k (mod 5) for some integer k . 4. Find a + b (mod n ), ab (mod n ), and ( a + b ) 2 (mod n ) if a = 4003, b =-127, and n = 85. 5. Find all integers 0 x < n satisfying each of the following congruences. (i) 3 x 4 (mod 6) (ii) 4 x 2 (mod 6) (iii) 4 x 3 (mod 7) (iv) 4 x 3 (mod 6) (v) 2 x 18 (mod 50) 1...
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This note was uploaded on 12/07/2011 for the course MATH 210 taught by Professor Staff during the Spring '08 term at University of Delaware.

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