I would start by saying that . Then if you multiply both sides of the congruence by 8, it becomes , and you can read off the answer for x directly from the index table.
Problem: Find all the solutions of the following congruence:
Since 5 is a primitive root of 23. I constructed an index table modulo 23 for the p.r. 5.
Number: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Index : 22, 2, 16, 4, 1, 18, 19, 6, 10, 3, 9, 20, 14, 21, 17, 8, 7, 12, 15, 5, 13, 11
Now, since can be rewritten in indices as
By looking at the index table, we get
Let ,
we get
Then, subtract 16 from both sides
From here I am stuck. I believe I am supposed to multiply 5 by some number to get 1 mod 22. From here I can solve for y and then for x.
Any help is greatly appreciated. Thank you.