Find integers x and y with 103x + 113y=1

Question

Find integers $x$ and $y$ with $103x + 113y=1$

How would you solve this problem? I'm thinking maybe you can use Euclidean Algorithm to solve for the inverse?

Answer 1

We can use the Extended Euclidean Algorithm. One implementation is the Euclid-Wallis Algorithm: $$ \begin{array}{r} &&1&10&3&3\\\hline 1&0&1&-10&31&-103\\ 0&1&-1&11&-34&113\\ 113&103&10&3&1&0\\ &&&&{\uparrow} \end{array} $$ The column with the arrow says that $$ 31\cdot113-34\cdot103=1 $$

Answer 2

Applying Gauss's algorithm allows us to solve it with simple mental arithmetic

$ 103x\!+\!113y=1\iff$ $ {\rm mod}\ 103\!:\ y\equiv \dfrac{1}{113}\equiv\dfrac{1}{10}\equiv\dfrac{10}{100}\equiv\dfrac{-93}{-3}\equiv 31$

Beware $\ $ Modular fraction arithmetic is well-defined only for fractions with denominator coprime to the modulus. See here for further discussion.

Answer 3

$$113y≡1(mod103) $$ $$10y≡1(mod103)$$ $$10y≡1+309(mod103)$$ $$y≡31(mod103)$$ Hence $y≡31(mod103).$
It is easy to show that $x≡-34(mod113).$