How do I solve modular arithmetic problems like 2x=7 mod(13)?

[XodoX]

How do I solve those problems?

Like,

Find some x such that x$$\equiv$$8 mod (18)

Find the inverse of 12 modulo 41

Solve 2x=7 mod (13)

I know it's easy, but I don't get it.

Let a and be be integers, and let m be a positive integer. Then a $$\equiv$$ b ( mod m) if and only if a mod m = b mod m

That's the explanation in the book. I'm not getting it. Can somebody please explain this modular arithmetic to me?

 

[XodoX]

Nobody knows? Maybe wrong forum.

 

[BWV]

mod arithmetic is just a system of integers that "starts" over at the mod #. Think about the 12 notes on a piano - c,c#,d,d# etc. once you get to b or 12 it just starts over. so 7 mod 13 could be (assuming you start at 1) 7, 20, 33, 46, etc

 

[bapowell]

Or hours on the clock -- they start over after 12. Compare military time with clock time: 20:00 hours is 8 o'clock because the clock time arithmetic is modular:

8 = 20 mod(12)