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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=653 and b=164737:

a	b	q	r	s1	s2	s3	t1	t2	t3
653	164737	0	653	1	0	1	0	1	0
164737	653	252	181	0	1	-252	1	0	1
653	181	3	110	1	-252	757	0	1	-3
181	110	1	71	-252	757	-1009	1	-3	4
110	71	1	39	757	-1009	1766	-3	4	-7
71	39	1	32	-1009	1766	-2775	4	-7	11
39	32	1	7	1766	-2775	4541	-7	11	-18
32	7	4	4	-2775	4541	-20939	11	-18	83
7	4	1	3	4541	-20939	25480	-18	83	-101
4	3	1	1	-20939	25480	-46419	83	-101	184
3	1	3	0	25480	-46419	164737	-101	184	-653

Answer

So we found that:

$gcd(653, 164737) = 1$
$s = -46419$
$t = 184$

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| = |-46419 × 653 + 184 × 164737| = |-30311607 + 30311608| = |1| = $ $1$
$gcd(a, b) = gcd(653, 164737) = $ $1$

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