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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=723 and b=914693:

a	b	q	r	s1	s2	s3	t1	t2	t3
723	914693	0	723	1	0	1	0	1	0
914693	723	1265	98	0	1	-1265	1	0	1
723	98	7	37	1	-1265	8856	0	1	-7
98	37	2	24	-1265	8856	-18977	1	-7	15
37	24	1	13	8856	-18977	27833	-7	15	-22
24	13	1	11	-18977	27833	-46810	15	-22	37
13	11	1	2	27833	-46810	74643	-22	37	-59
11	2	5	1	-46810	74643	-420025	37	-59	332
2	1	2	0	74643	-420025	914693	-59	332	-723

Answer

So we found that:

$gcd(723, 914693) = 1$
$s = -420025$
$t = 332$

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| = |-420025 × 723 + 332 × 914693| = |-303678075 + 303678076| = |1| = $ $1$
$gcd(a, b) = gcd(723, 914693) = $ $1$

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