Final answer is enough, will upvote

6. 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}

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}

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}

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}

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}

7. The prime factorization of 198 is

198 = 2 3² 11

Group of answer choices

2 3.3 11

2.33.3

1.198

18.11

8. Some numbers and their prime factorizations are given below.

140 = 2² 5 7

175 = 5² 7

532 = 2² 7 19

648 = 2³ 3⁴

1078 = 2 7² 11

1083 = 3 19²

15435 = 3² 5 7³

1078 = 2 7² 11

25480 = 2³ 5 7²13

Using these prime factorizations the gcd(532, 15435) =

7

52

97

77

9.Applying the Euclidean Algorithm and the Extended Euclidean Algorithm.

the gcd(81,60) , and expression of the gcd as a linear combination of the two numbers =

gcd=3, combination = 3.81-4.60

gcd=3, combination = 21-18

gcd=12, combination = 81-69

gcd=9, combinaton = 81-60-12

10. using arithmetic in the ring of integers mod 9,

58inZ9=

4

36

2

0