What is the formula for (m-1)^-1 and how can it be proven?

[sarah77]

Homework Statement

Find a formula for (m-1)^-1 and prove that your result holds true in general.

Homework Equations

if m=5: (5-1) in Z5 is 4 and inverse of 4 in Z5 is 4.
m=6: (6-1) in Z6 is 5 and inverse of 5 in Z6 is 5.
and so on.

The Attempt at a Solution

I found the formula: (m-1)^-1 = (m-1) in Zm, but I do not know how to prove it..please help!

 

[lanedance]

the problem statement is a little vague...

though i am guessing you are trying to find a generic formula, in terms of m, for the inverse element for (m-1) in multiplicative group of integers modulo m?

 

[Dick]

If you think (m-1) is the inverse of (m-1) in Z_m, then you could prove it by showing (m-1)(m-1) is congruent to 1 mod m.

 

[sarah77]

I apologize, the question reads: Find the multiplicative inverse of m-1 in Zm for several values of m. Find a formula for (m-1)^-1 and prove that your result holds in general. How could I use (m-1)(m-1)?

 

[Dick]

I apologize, the question reads: Find the multiplicative inverse of m-1 in Zm for several values of m. Find a formula for (m-1)^-1 and prove that your result holds in general. How could I use (m-1)(m-1)?

Multiply it out. Can you show its remainder when divided by m is 1?

 

[sarah77]

OH! Ok, thank you!