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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=764 and b=422179:

a	b	q	r	s1	s2	s3	t1	t2	t3
764	422179	0	764	1	0	1	0	1	0
422179	764	552	451	0	1	-552	1	0	1
764	451	1	313	1	-552	553	0	1	-1
451	313	1	138	-552	553	-1105	1	-1	2
313	138	2	37	553	-1105	2763	-1	2	-5
138	37	3	27	-1105	2763	-9394	2	-5	17
37	27	1	10	2763	-9394	12157	-5	17	-22
27	10	2	7	-9394	12157	-33708	17	-22	61
10	7	1	3	12157	-33708	45865	-22	61	-83
7	3	2	1	-33708	45865	-125438	61	-83	227
3	1	3	0	45865	-125438	422179	-83	227	-764

Answer

So we found that:

gcd(764, 422179) = 1
s = -125438
t = 227

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| = |-125438 × 764 + 227 × 422179| = |-95834632 + 95834633| = |1| = 1
gcd(a, b) = gcd(764, 422179) = 1

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