I need help in solving the equation:
x^25=2 (mod 133)
Thank's in advance.
Suppose and . Then, by the Chinese Remainder Theorem, . Indeed, . This is a solution (so the solutions 14 and 3 of the second equation are not incorrect... at least 3 is correct).
Suppose . Then, , and , so this is not a solution.
Suppose . Then, , and , so again, this is not a solution.
Hence, the only solution is .
Use an online Chinese Remainder Theorem Calculator? You need familiarity with the subject to be able to calculate solutions otherwise.
Here is a link to a calculator that will do it for you: Chinese Remainder Theorem Calculator
How it works: Suppose
Then, we know for some integer . So, the possible equivalence classes of are . We check each and discover that only .
Do the same for or .