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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. Note that you need to enter n before b.
E.g. if you want to know the multiplicative inverse of 26 mod 11, then use n=11 and b=26.

Output

Note: 798 mod 661 ≡ 137, so you could also use this calculator with n=661 and b=137. In that case the table will have less rows, even though it has the same multiplicative inverse.

This is the calculation for finding the multiplicative inverse of 798 mod 661 using the Extended Euclidean Algorithm:

n	b	q	r	t1	t2	t3
661	798	0	661	0	1	0
798	661	1	137	1	0	1
661	137	4	113	0	1	-4
137	113	1	24	1	-4	5
113	24	4	17	-4	5	-24
24	17	1	7	5	-24	29
17	7	2	3	-24	29	-82
7	3	2	1	29	-82	193
3	1	3	0	-82	193	-661

Answer

So t = 193. Now we still have to apply mod n to that number:
193 mod 661 ≡ 193
So the multiplicative inverse of 798 modulo 661 is 193.

Verification

Let i be the answer we just found, so i=193. We also have b=798 and n=661.
If our answer is correct, then i × b (mod n) ≡ 1 (mod n).
Let's see if that's indeed the case.
i × b (mod n) ≡
193 × 798 (mod 661) ≡
154014 (mod 661) ≡
1 (mod 661)

As you can see, i × b (mod n) ≡ 1 (mod n), so our calculation is correct!