Explanation Proof: We are given that a c ≡ b c ( mod n ) and gcd ( c , n ) = 1 . We have to show that a ≡ b ( mod n ) . As gcd ( c , n ) = 1 , we have c x + n y = 1 for integers x and y . ⇒ c x = 1 − n y Since, a c ≡ b c ( mod n ) ⇒ a c − b c = n k for some integer k . Multiplying the equation by x , we have ( a − b ) c x = n k x Substituting the value of c x , we have

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ISBN: 9780131679955

Author: Edgar G. Goodaire

Publisher: Prentice Hall

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Chapter 4.4, Problem 12E

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To prove: That a≡b(modn) if ac≡bc(modn) and gcd(c,n)=1.

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Chapter 4.4, Problem 11E

Chapter 4.4, Problem 13E

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Chapter 4 Solutions

Discrete Mathematics with Graph Theory

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Ch. 4.1 - Prob. 1E Ch. 4.1 - 2. True or false? If false, give a counterexample....Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - 9. Suppose is a natural number and is the...Ch. 4.1 - Prob. 10E Ch. 4.1 - Prob. 11E Ch. 4.1 - Prob. 12E Ch. 4.2 - True/False Questions Given two consecutive...Ch. 4.2 - Prob. 2TFQ Ch. 4.2 - Prob. 3TFQ Ch. 4.2 - Prob. 4TFQ Ch. 4.2 - Prob. 5TFQ Ch. 4.2 - Prob. 6TFQ Ch. 4.2 - Prob. 7TFQ Ch. 4.2 - Prob. 8TFQ Ch. 4.2 - Prob. 9TFQ Ch. 4.2 - Prob. 10TFQ Ch. 4.2 - Prob. 1E Ch. 4.2 - Prob. 2E Ch. 4.2 - Draw the Hasse diagrams for each of the following...Ch. 4.2 - List all minimal, minimum, maximal, and maximum...Ch. 4.2 - Let n be a natural number. 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To prove: That a ≡ b ( mod n ) if a c ≡ b c ( mod n ) and gcd ( c , n ) = 1 .

Solution Summary: The author proves that aequiv b(mathrmmodn) has cx+ny=1 for integers.

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To prove: That a≡b(modn) if ac≡bc(modn) and gcd(c,n)=1.

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Chapter 4 Solutions