Hi, I am currently stuck on an Algebra question regaarding the table of powers, I have a link below which will take you to a screenshot that I hav took of the question, I have currently done part a and I am not understanding how you can use the table of powers [2]^k to calculate powers of [3]^k. Any help would be appreciated. Thanks. 
http://prntscr.com/98yxgp
10.3 In this question, work in \(\displaystyle \, \mathbb{Z}/13\mathbb{Z}.\,\) Classes [a]13 will be abbreviated as [a].
(a) Compute and make a table of the powers [2]k for k = 1, 2, ..., 12 (say, with two columns, or two rows, if you prefer, with the values of k in the first and the values of [2]k in the second).
(b) Use the table you have found in part (a) to compute the first few powers of [3], and find the smallest positive integer k such that [3]k = 1.
(c) Use the table you have found in part (a) to compute the first few powers of [4], and find the smallest positive integer k such that [4]k = 1.
(d) Use the table you have found in part (a) to compute the first few powers of [5], and find the smallest positive integer k such that [5]k = 1.
(e) Use the table you have found in part (a) to compute the first few powers of [6], and find the smallest positive integer k such that [6]k = 1.
Note: If you wish, in parts (b), (c), (d), and (e), you may compute the powers of those classes directly as a check for yourself, but in the submitted solution you must obtain them (explaining briefly how) from the table of powers of [2] which you have calculated in part (a).
http://prntscr.com/98yxgp
10.3 In this question, work in \(\displaystyle \, \mathbb{Z}/13\mathbb{Z}.\,\) Classes [a]13 will be abbreviated as [a].
(a) Compute and make a table of the powers [2]k for k = 1, 2, ..., 12 (say, with two columns, or two rows, if you prefer, with the values of k in the first and the values of [2]k in the second).
(b) Use the table you have found in part (a) to compute the first few powers of [3], and find the smallest positive integer k such that [3]k = 1.
(c) Use the table you have found in part (a) to compute the first few powers of [4], and find the smallest positive integer k such that [4]k = 1.
(d) Use the table you have found in part (a) to compute the first few powers of [5], and find the smallest positive integer k such that [5]k = 1.
(e) Use the table you have found in part (a) to compute the first few powers of [6], and find the smallest positive integer k such that [6]k = 1.
Note: If you wish, in parts (b), (c), (d), and (e), you may compute the powers of those classes directly as a check for yourself, but in the submitted solution you must obtain them (explaining briefly how) from the table of powers of [2] which you have calculated in part (a).
Last edited by a moderator: