The two ways I know of are: (1) Extended Euclidean algorithm (2) Modular exponentiation. See here for some discussion. If you'd like an example, just ask, and I or someone else will provide one.
There is a general method to find the inverse , you know , it is Euclidean Algorithm , but i promise once you have practised a lot , it would be very easy for you to solve , without Euclidian Algorithm .
Also, without going too deeply in the number theoretical realm, if you know the Euler totient of the modulus, you can efficiently compute :
In order to do this you use the modular exponentiation (square and multiply) algorithm, and is the Euler totient of 23 (which incidentally equals 22).
Hello, Migotek84!
Here's a very primitive method . . .
Compute: .
Let: .
Multiply by 17: .
Then: .
.[1]
Since is an integer, is a multiple of 17.
. .
.[2]
Since is an integer, is a multiple of 6.
. .
.[3]
Since is an integer, is a multiple of 5.
. . The first time this happens is when .
Substitute into [3]: .
Substitute into [2]: .
Substitute into [1]: .
Therefore: .