Question

How do you solve 10x + 5 $\equiv$ 2mod(13)? I know how to solve this without the + 5 on the left hand side. But how does the approach to the problem change when you have addition or subtraction on the left side?

Answer 1

10x + 5 = 2 mod(13)

=>  10x + 5 - 5 = 2 mod(13) - 5
=>  10x = 2 mod(13) - 5 + 13    (we can add 13, because it is mod 13)
=>  10x = 2 mod(13) + 8
=>  10x = 10 mod(13)
=>  so, x = 1.

Answer 2

10x + 5 = 2 mod(13)

=>  10x + 5 - 5 = 2 mod(13) - 5
=>  10x = 2 mod(13) - 5 + 13    (we can add 13, because it is mod 13)
=>  10x = 2 mod(13) + 8
=>  10x = 10 mod(13)
=>  so, x = 1.