Calculator
For the Euclidean Algorithm, Extended Euclidean Algorithm and multiplicative inverse.
Before you use this calculator
If you're used to a different notation, the output of the calculator might confuse you at first.
Even though this is basically the same as the notation you expect. If that happens, don't panic.
Just make sure to have a look the following pages first and then it will all make sense:
- Euclidean Algorithm
For the basics and the table notation - Extended Euclidean Algorithm
Unless you only want to use this calculator for the basic Euclidean Algorithm. - Modular multiplicative inverse
in case you are interested in calculating the modular multiplicative inverse of a number modulo n
using the Extended Euclidean Algorithm
Input
AlgorithmChoose which algorithm you would like to use.
Numbers
Enter the input numbers. Note that you need to enter n before b.
E.g. if you want to know the multiplicative inverse of 26 mod 11, then use n=11 and b=26.
Output
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 665 | 145981 | 0 | 665 | 0 | 1 | 0 |
| 145981 | 665 | 219 | 346 | 1 | 0 | 1 |
| 665 | 346 | 1 | 319 | 0 | 1 | -1 |
| 346 | 319 | 1 | 27 | 1 | -1 | 2 |
| 319 | 27 | 11 | 22 | -1 | 2 | -23 |
| 27 | 22 | 1 | 5 | 2 | -23 | 25 |
| 22 | 5 | 4 | 2 | -23 | 25 | -123 |
| 5 | 2 | 2 | 1 | 25 | -123 | 271 |
| 2 | 1 | 2 | 0 | -123 | 271 | -665 |
So $t = 271$. Now we still have to apply $\pmod{n}$ to that number: $$271 \pmod{665} ≡ 271$$ So the multiplicative inverse of $145981$ modulo $665$ is $271$.
VerificationLet $t$ be the answer we just found, so $t=271$. We also have $b=145981$ and $n=665$.
If our answer is correct, then $t \times b \pmod{n} \equiv 1 \pmod{n}$.
Let's see if that's indeed the case.
\[
\begin{aligned}
t \times b \pmod{n} &\equiv \\
271 \times 145981 \pmod{665} &\equiv \\
39560851 \pmod{665} &\equiv1 \pmod{665} \\&\equiv 1 \pmod{n}\end{aligned}
\]