4*4= 16= 1 (mod 5) so any number of the form 5j+ 4 satisfies 4x= 1 (mod 5). -x+ 2= 3 (mod 5) leads to -x= 1 (mod 5) and so x= -1= 4 (mod 5). Any number of the form 5j+ 4 also satisfies this. Of course, to call 4 a "unique" solution you have to call two numbers that are equal "mod 5" the same: 4, 9, 14, 19, etc. all satify those two equations but are the same "mod 5".

No. 7(1)= 7 which is NOT equal to 1 (mod 8). -1+ 2= 1 which is NOT equal to 3 (mod 8). 7*7= 49= 1+ 6(8)= 1 (mod 8) so the any number of the form 7+ 8n satisfies 7x= 1 (mod 8). -x+ 2= 3 (mod 8) leads to -x= 1 (mod 8) so x= -1= 7 (mod 8).Also...

-x + 2 = 3 (mod 8)

7x = 1 (mod 8)

Again there is a unique solution, namely x = 1 (mod 8)

Are you asking why the solutions are unique? That is not unusual. Typically linear equations have only one solution. What is unusual is that thereCan anyone explain where these solutions come from? I understand how to get to 4x = 1 (mod 5) and 7x = 1 (mod 8) but I dont get how we get the 'unique solutions' at the end? I'd appreciate a clear explanation please. Thank youexistin each case, a number that satisfies both equations. And that is true because the two equations reduce to the same thing.