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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: 36533 mod 419 ≡ 80, so you could also use this calculator with n=419 and b=80. 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 36533 mod 419 using the Extended Euclidean Algorithm:

n	b	q	r	t1	t2	t3
419	36533	0	419	0	1	0
36533	419	87	80	1	0	1
419	80	5	19	0	1	-5
80	19	4	4	1	-5	21
19	4	4	3	-5	21	-89
4	3	1	1	21	-89	110
3	1	3	0	-89	110	-419

Answer

So t = 110. Now we still have to apply mod n to that number:
110 mod 419 ≡ 110
So the multiplicative inverse of 36533 modulo 419 is 110.

Verification

Let i be the answer we just found, so i=110. We also have b=36533 and n=419.
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) ≡
110 × 36533 (mod 419) ≡
4018630 (mod 419) ≡
1 (mod 419)

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