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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

Algorithm

Choose which algorithm you would like to use.

Numbers

Enter the input numbers:

Output

This is the output of the Extended Euclidean Algorithm using the numbers a=991 and b=667329:

a	b	q	r	s1	s2	s3	t1	t2	t3
991	667329	0	991	1	0	1	0	1	0
667329	991	673	386	0	1	-673	1	0	1
991	386	2	219	1	-673	1347	0	1	-2
386	219	1	167	-673	1347	-2020	1	-2	3
219	167	1	52	1347	-2020	3367	-2	3	-5
167	52	3	11	-2020	3367	-12121	3	-5	18
52	11	4	8	3367	-12121	51851	-5	18	-77
11	8	1	3	-12121	51851	-63972	18	-77	95
8	3	2	2	51851	-63972	179795	-77	95	-267
3	2	1	1	-63972	179795	-243767	95	-267	362
2	1	2	0	179795	-243767	667329	-267	362	-991

Answer

So we found that:

$gcd(991, 667329) = 1$
$s = -243767$
$t = 362$

Verification
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| = |-243767 × 991 + 362 × 667329| = |-241573097 + 241573098| = |1| = $ $1$
$gcd(a, b) = gcd(991, 667329) = $ $1$

As you can see, $| s × a + t × b| = gcd(a, b)$, so our calculation is correct!