Since the multiplier  " 5 " and the module "8" are mutually prime numbers,
 
the general theory says that this modular equation has the solution, and the solution is unique as the number module 8.


The method to find the solution is  "trial and error".


Do not afraid and do not be scared. To find the solution, you need to check only 8 numbers module 8.

These numbers to check are 0 mod8, 1 mod8, 2 mod8, 3 mod8, 4 mod8, 5 mod8, 6 mod8 and 7 mod8.


The solutions INEVITABLY  is among these numbers.


Doing in this way, you will quickly find that x= 5 mod8 is the solution.


Indeed, 5*5 = 25 = 1 mod8.


It means that the solutions are all integer numbers that give the remainder 5 when are divided by 8.


In other words, all integer numbers of the form  n = 5 + 8k,  k = 0, +/-1, +/-2, . . .