- #1

FritoTaco

- 133

- 23

## Homework Statement

Hi, I'm doing a problem by solving congruences but my question is simply trying to find the inverse of [itex]2 \enspace (mod\enspace 17)[/itex] from [itex] 2x \equiv 7(mod \enspace 17)[/itex].

## Homework Equations

It's hard to find a definition that makes sense but if you check my upload images you can see exactly what I'm trying to do.

Source:

## The Attempt at a Solution

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__Going forwards__

1.) [itex]2 = 17(0) + 2[/itex]

2.) [itex]17 = 2(8) + 1[/itex]

3.) [itex]2 = 1(2) + 0[/itex]

Our gcd = 1 which means we can find the inverse starting with step 2.

__Going backwards__

4.) [itex]1 = 17 - 2(8)[/itex]

5.) [itex]2 = 2 - 17(0)[/itex] we wrote this from step 1 by setting the remainder equal to everything else.

6.) [itex]1 = 17 - (2 - 17(0))(8)[/itex] plug in the equation from step 5 into [itex]2[/itex] from step 4.

7.) [itex]1 = 17 - 2(8)[/itex]

So our answer says the inverse is 8, but the answer is 9. Where am I messing up?