Solving Linear Congruence Equation: Finding the Nonnegative Integer Representation

Question

I have a question for a part of the following problem: Solve the linear congruence 7x ≡ 6(mod 29)

I understand how to find the linear combination equality using the extended Euclidean Algorithm, which is this: 1 = 1⋅29 − 4⋅7

But what is throwing me is that I can't find a way to find the nonnegative integer representation less than 29 of the inverse of 7. I know by my equation that -4 is the inverse of 7, but how would I go about this?

Answer 1

It is written down explicitly in your post. You wrote that using the Extended Euclidean Algorithm, you reached $1=1\cdot 29-4\cdot 7$. This says that $-4$ is the inverse of $7$. If you want positive, use $29+(-4)$.

Answer 2

Multiply $7x\equiv 6$ by $-4$ on both sides to get $x\equiv -24\equiv 5$.

Sorry, misread the question. $-4\equiv 25\pmod{29}$, so $25$ is the least positive representation of $-4$.