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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=919 and b=430363:

a	b	q	r	s1	s2	s3	t1	t2	t3
919	430363	0	919	1	0	1	0	1	0
430363	919	468	271	0	1	-468	1	0	1
919	271	3	106	1	-468	1405	0	1	-3
271	106	2	59	-468	1405	-3278	1	-3	7
106	59	1	47	1405	-3278	4683	-3	7	-10
59	47	1	12	-3278	4683	-7961	7	-10	17
47	12	3	11	4683	-7961	28566	-10	17	-61
12	11	1	1	-7961	28566	-36527	17	-61	78
11	1	11	0	28566	-36527	430363	-61	78	-919

Answer

So we found that:

$gcd(919, 430363) = 1$
$s = -36527$
$t = 78$

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| = |-36527 × 919 + 78 × 430363| = |-33568313 + 33568314| = |1| = $ $1$
$gcd(a, b) = gcd(919, 430363) = $ $1$

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