Answered: I. Solve for x. All final answers must…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.5: Congruence Of Integers

Problem 4TFE: Label each of the following statements as either true or false. a is congruent to b modulo n if and...

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I. Solve for x. All final answers must be in Znwhere nis the modulus.
(a) x≡3(15) + 45 mod 18
(b) x≡−12 mod 13
(c) 6xmod 11 ≡1
(d) x+ 2 mod 9 ≡0

II. Write down the congruence classes formed, modulo 4.

III. Find a multiplicative inverse of 3 modulo 6, if there are any.

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a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a0-a1+a2-+(1)nan, when z is written in the form as described in the previous problem. a. Prove that 10n1(mod9) for every positive integer n. b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. (Hint: Any integer can be expressed in the form an10n+an110n1++a110+a0 where each ai is one of the digits 0,1,...,9.)
Label each of the following statements as either true or false. a is congruent to b modulo n if and only if a and b yield the same remainder when each is divided by n.
30. Prove statement of Theorem : for all integers .

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I. Solve for x. All final answers must be in Znwhere nis the modulus. (a) x≡3(15) + 45 mod 18 (b) x≡−12 mod 13 (c) 6xmod 11 ≡1 (d) x+ 2 mod 9 ≡0 II. Write down the congruence classes formed, modulo 4. III. Find a multiplicative inverse of 3 modulo 6, if there are any.