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

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Homework Help Overview

The discussion revolves around finding a formula for the multiplicative inverse of (m-1) in the context of modular arithmetic, specifically within the group of integers modulo m.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants explore the idea of proving that (m-1) is its own inverse in Z_m by examining the congruence relation (m-1)(m-1) ≡ 1 (mod m). There is also a focus on finding a general formula for the inverse in terms of m.

Discussion Status

Some participants are clarifying the problem statement and attempting to establish a proof for the proposed formula. There is an ongoing exploration of how to demonstrate the relationship between (m-1) and its inverse, with various interpretations being discussed.

Contextual Notes

Participants note the need to prove the result holds true in general and are considering specific values of m to illustrate their points. The original poster's request for help indicates a lack of clarity on how to proceed with the proof.

sarah77
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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!
 
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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?
 


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.
 


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)?
 


sarah77 said:
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?
 


OH! Ok, thank you!
 

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