hw04 - a, · · · , ( p − 1) · a are congruent modulo p...

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In the Name of God Fall 2005 Discrete Mathematics Homework 4 Due: Aban 15 Problem 1. Let p be an odd prime number, let a be any integer, and let b = a ( p - 1) / 2 . Show that b mod p is either 0 or 1 or p 1. ( Hint: Consider ( b + 1)( b 1) . ) Problem 2. If n, m are integers, evaluate: (a) n + m 2 + n - m +1 2 . (b) n + m 2 + n - m +1 2 . (The special case m = 0 is worth nothing.) Problem 3. If n > 1 is an integer, Then prove that n 4 +4 n is never a prime. Problem 4. Prove that if n is an integer, (a) 2 n 1 is not a power of 3, where n > 2. (a) 2 n + 1 is not a power of 3, where n > 3. Problem 5. Show that among ±ve integers, there are always three with sum divisible by 3. Problem 6. This problem outlines a proof of Fermat’s Little Theorem. (a) Suppose that a is not divisible by the prime p . Show that no two of the integers 1 · a, 2 ·
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Unformatted text preview: a, · · · , ( p − 1) · a are congruent modulo p . (b) Conclude from part (a) that the product of 1 , 2 , · · · , p − 1 is congruent 1 modulo p to the product of a, 2 a, · · · , ( p − 1) a . Use this to show that ( p − 1)! ≡ a p-1 ( p − 1)! ( mod p ) (c) Use Wilson’s theorem to show that a p-1 ≡ 1 (mod p ) if a is not divisible by p . (d) Use part (c) to show that a p ≡ a (mod p ) for all integers a . Problem 7. Complete the proof of the Chinese Remainder Theorem by showing that the simultaneous solution of a system of linear congruences modulo pairwise relatively prime integers is unique modulo the product of these moduli. 2...
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This note was uploaded on 02/06/2011 for the course ECE 423 taught by Professor Dolatabadi during the Spring '11 term at Amirkabir University of Technology.

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hw04 - a, · · · , ( p − 1) · a are congruent modulo p...

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