Please answer question 11 811 For each of the following cong

8-11. For each of the following congruences, determine whether there erists a positive integer k so that the congruence is satisfied. If so, find the smallest such k. 9. 3 s 1 (mod 17) 8. 2 1 (mod 11) 11. 4 s 1 (mod 19) 10. 2 1 (mod 14)

Solution

8) 2^k = 1 mod 11

By trial and error we find that

1024-1 = 1023 is divisible by 11

Hence k = 10

9) 3^k = 1 mod 17

When k =16

3^k = 43046720

3^k-1 = 2532160 (17)

Hence k =16 is the least positive solution

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10) 2^k = 1 mod 14

Since 2^k -1 is odd and 14 is even no solution

11) 4^k = 1 mod 19

4^9 =262144

262143 = 19(13797)

Hence k =9 is answer.