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.
NumbersEnter the input numbers:
Output
This is the output of the Extended Euclidean Algorithm using the numbers a=689 and b=107:| a | b | q | r | s1 | s2 | s3 | t1 | t2 | t3 |
|---|---|---|---|---|---|---|---|---|---|
| 689 | 107 | 6 | 47 | 1 | 0 | 1 | 0 | 1 | -6 |
| 107 | 47 | 2 | 13 | 0 | 1 | -2 | 1 | -6 | 13 |
| 47 | 13 | 3 | 8 | 1 | -2 | 7 | -6 | 13 | -45 |
| 13 | 8 | 1 | 5 | -2 | 7 | -9 | 13 | -45 | 58 |
| 8 | 5 | 1 | 3 | 7 | -9 | 16 | -45 | 58 | -103 |
| 5 | 3 | 1 | 2 | -9 | 16 | -25 | 58 | -103 | 161 |
| 3 | 2 | 1 | 1 | 16 | -25 | 41 | -103 | 161 | -264 |
| 2 | 1 | 2 | 0 | -25 | 41 | -107 | 161 | -264 | 689 |
So we found that:
- $gcd(689, 107) = 1$
- $s = 41$
- $t = -264$
If our answer is correct, then the absolute value of $s × a + t × b$ is equal to the $gcd$ of $a$ and $b$. We have:
- $|s × a + t × b| = |41 × 689 + -264 × 107| = |28249 + -28248| = |1| = $ $1$
- $gcd(a, b) = gcd(689, 107) = $ $1$
As you can see, $| s × a + t × b| = gcd(a, b)$, so our calculation is correct!