# Problems with inverses in arithmetic in ring z

## Homework Statement

Calculate 7*11 + 9*11^-1 in the group Z20

## The Attempt at a Solution

77+ (9*1/11) in group Z20
77 + 9/11
17 +11x= 20mod+9
My solution was 12, this makes 149 on both sides when you multiply the mod times 7.
I am doing independent study and the computer program I am using says the answer is 16.
What am I doing wrong? Btw, what is the difference between ring congruence arithmetic and regular congruence arithmetic? They seem pretty similar.

## The Attempt at a Solution

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Dick
Homework Helper
I'm not really sure what you are doing. What is 20mod+9 supposed to mean? At some point you need to figure out what 11^(-1) mod 20 is. Why not do it first?

Dick
Homework Helper
bump
You can 'bump' a post if no one has given any response. I don't think that's the case here.

The first thing you absolutely need to do is to figure out what 11^-1 mod 20 is.

The first thing you absolutely need to do is to figure out what 11^-1 mod 20 is
original problem: calculate 7*11+9*11^-1 with mod 20

So in order to figure out the inverse of 11^-1 mod 20, I would do the typical
11x=mod 20+1, and from here would I go 17+99x=mod 20+1(I get 16 on this one, the supposedly correct solution), or some other route?

HallsofIvy
Homework Helper
To find the multiplicative inverse of 11 mod 20, you want a number, m, such that 11m= 1 (mod 20) or 11m= 1+ 20n for some integer n.

That is the same as solving the diophantine equation 11m- 20n= 1.

11 divides into 20 once with remainder 9: 20- 11= 9.

9 divides into 11 once with remainder 2: 11- 9= 2.

2 divides into 9 four times with remainder 1: 9- 4(2)= 1.

Replace the "2" in that equation with 11- 9: 9- 4(11- 9)= 5(9)- 4(11)= 1.

Replace the "9" in that equation with 20- 11: 5(20- 11)- 4(11)= 5(20)- 9(11)= 1.
(Of course: 100- 99= 1.)

A solution to 11m- 20n= 1 is m= -9, n= -5. Thus, the multiplicative inverse of 11, mod 20, is -9= 20- 9= 11 (mod 20). That is, the multiplicative inverse of 11, mod 20, is 11 itself.

Check: 11(11)= 121= 6(20)+ 1= 1 (mod 20).