Euclidean Algorithm for Modular Inverse, with negative numbers I might be on to something quite simple which I'm failing to see, while calculating modular inverses.
For example, calculating 7x = 5 (mod 12)
Which is the same as saying: 7x - 5 = 12k
Which becomes: 7x - 12k = 5
And then I proceed using Euclidean Algorithm for x,k. I get to -25 and 15 respectively. However, I need the x to be positive to get the inverse I'm looking for. How can I get a positive modular inverse?
Thanks in advance!

In a Bezout identity
$$
a⋅x+b⋅y=c
$$
you can exchange multiples of $a⋅b$ or even $lcm(a,b)=a'\cdot b=a\cdot b'$ between the terms on the left, so that
$$
a⋅(x-k⋅b')+b⋅(y+k⋅a')=c
$$
is also a correct identity. This slightly extends the reasoning on modular equivalences in the comment of  Bill Dubuque.
