Hello oldguynewstudentYou're correct in what you have so far. Let me summarise:

The problem is to find the value(s) of for which ; in other words the value(s) of for which:, where is another integerYou have correctly found, using the Euclidean Algorithm, that

So you have the solution

So you have, in fact, found an inverse of , and that is . Check it out:

But perhaps you were looking for an inverse between and . So you simply add enough 's to your solution until you get a value in this range. This works because

for any integerOf course, all you need to add is itself, and get . So is also an inverse of . Check this one out:

This gives us another solution to the equation

and that is

There are infinitely many others that you can find in a similar way: ...etc.

Does that help to clear things up?

Grandad