Anyone know modular (modulus) arithmetic?

[Avalon]

So I had trouble with a homework problem a week ago, and my professor gave me the solution to it for study purposes...however, this also confuses me. Here's the little problem.

Find the modular inverse of f(x) = 17x + 4

f(x) = 17x + 4 (mod 26), so we can set up solving for the inverse function as follows:

x = 17f-1(x) +4 (mod 26)
23*17f-1(x) = 23*(x ? 4) (mod 26), because 17-1 = 23 (mod 26)
f-1(x) = 23*(x ? 4) (mod 26)
f-1(x) = 23x ? 92 (mod 26)
f-1(x) = 23x + 12 (mod 26)

Now, I understand that in mod26, every number 0-25 has a modular inverse...wish I had the list. Apparently 17 and 23 are modular inverses, as shown in this solution. So I understand that. And I understand how he multiplied the function by 23 to cancel out the 17 and get f-1(x) by itself. Finally understood all of this, except how he got from the second to last line to the last line. Why did the -92 turn into +12?

 

[chuckywang]

-92 (mod 26) = -92+4*26 (mod 26) = 12 (mod 26)

And BTW, not every number 0-25 has an inverse mod 26. For example, 13 has no inverse mod 26.

 

[Avalon]

Oh, so you just multiply it by a multiple of 26 until you reach a positive number?

 

[chuckywang]

Originally posted by: Avalon
Oh, so you just multiply it by a multiple of 26 until you reach a positive number?

Yep, pretty much. 26 = 0 mod 26 obviously, so you can add/subtract as many 26's as you want without affecting the equality.

 

[Avalon]

Awesome. I'm looking back at my old homework stuff and my notes from class that day and all the problems make perfect sense now. Thanks a ton!