# Finding possible inverses of a modulo function

I know how to find $one$ inverse via the euclidean algorithm, but I can't figure out how to find more of them.

For example:

Find an inverse $x$, of $57$ $modulo$ $100$ Or an $x$ such that $57x ≡ 1$ modulo 100

I got the answer $-7$ from the euclidean algorithm, but then the domain of x is restricted to be between $0$ and $100$. I know $93$ works, but not how I would go about finding that on paper.

Thanks

• Hint: Any two inverses will be congruent mod $100$. – Tobias Kildetoft Apr 27 '15 at 17:55
• Hint 93 = 100-7 – Surb Apr 27 '15 at 17:56
• you can write $\frac{1}{57} and adding or subtracting 100 to the numerator until we get a integer number – Dr. Sonnhard Graubner Apr 27 '15 at 17:59 ## 1 Answer Once you get one answer$x=-7$then all the others can be obtained by using the fact that$x \equiv -7 \pmod{100}$. Thus every such$x$is of the form$x=-7+100t$, where$t\in \mathbb{Z}$. • errrm...$\bmod 100\$ cough – AlexR Apr 27 '15 at 17:57
• @AlexR thanks for pointing the typo. – Anurag A Apr 27 '15 at 17:58
• Man that was really simple. I feel dumb now haha. Thanks for the help. – Avernikas Apr 27 '15 at 18:03