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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: 851 mod 653 ≡ 198, so you could also use this calculator with n=653 and b=198. 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 851 mod 653 using the Extended Euclidean Algorithm:

n	b	q	r	t1	t2	t3
653	851	0	653	0	1	0
851	653	1	198	1	0	1
653	198	3	59	0	1	-3
198	59	3	21	1	-3	10
59	21	2	17	-3	10	-23
21	17	1	4	10	-23	33
17	4	4	1	-23	33	-155
4	1	4	0	33	-155	653

Answer

So t = -155. Now we still have to apply mod n to that number:
-155 mod 653 ≡ 498
So the multiplicative inverse of 851 modulo 653 is 498.

Verification

Let i be the answer we just found, so i=498. We also have b=851 and n=653.
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) ≡
498 × 851 (mod 653) ≡
423798 (mod 653) ≡
1 (mod 653)

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