Answered: For all integers a, b, and c, if a | b…

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 51E: In the congruences ax b (mod n) in Exercises 40-53, a and n may not be relatively prime. Use the...

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Question

1): For all integers a, b, and c, if a | b or a | c, then a | bc

A): True

b): False

2): Compute the value of the following expressions without a calculator using the Congruence Modulo Laws

46³⁰ mod 9

A): 1

B): 0

C): 67390

D): too large to compute

3): Group the following numbers according to congruence mod 13.

That is, put two numbers in the same group if they are equivalent mod 13.

{−63, -54, -41, 11, 13, 76, 80, 130, 132, 137}

A): Congruent to 0 mod 13: {13, 130} Congruent to 2 mod 13: {-63, 80, 132 }Congruent to 7 mod 13: {137 }Congruent to 11 mod 13: {-54, -41, 11, 76}

b): Congruent to 0 mod 13: {13, 130} Congruent to 2 mod 13: {80, 132 }Congruent to 7 mod 13: {137 }Congruent to 11 mod 13: { 11, 76} Congruent to -7 mod 13 {-63} Congruent to 3 mod 13 {-41}

C): Congruent to 0 mod 13: {13, 130} Congruent to 2 mod 13: {-63, 80, 132 }Congruent to 7 mod 13: {137 } Congruent to -2 Mod 13 {-54,=41} Congruent to 11 mod 13: { 11, 76}

D): Congruent to 0 mod 13: {13, 130} Congruent to 2 mod 13: {-54 80, 132 }Congruent to 7 mod 13: {137 }Congruent to 11 mod 13: {-63, -41, 11, 76}

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