What must hold true in order fro?

Question

For example

a) 1653x ≡ 77(mod 2000)
b) x^2 ≡ 4x (mod 12)

which ones of these have solutions sand why...and how do you know what the complete solution is

thanks

Answer 1

a.No Solutions

To see this, go to Excel. In column 1, put all the numbers from 0 through 1999. In column 2, multiply the first number by 1653. In the next column, use the Mod function to see the result Then verify that there is no solution

b. Solutions are when x=0, x=4, x=6 or x=10

8 is the only even number not a solution.

It has been a long time since I studied this stuff. I seem to remember there was some rule by Euler for finding the answers more quickly.