# Modulo arithmetic

• Dec 1st 2007, 03:07 PM
Joel24
Modulo arithmetic
Hey, just stuck on this question:

Show that in $Z_m$ we have $(m-1)^{-1} = m-1$

I have no idea. Please help! $Z$ is used to represent the set of integers.
• Dec 2nd 2007, 04:49 AM
topsquark
Quote:

Originally Posted by Joel24
Hey, just stuck on this question:

Show that in $Z_m$ we have $(m-1)^{-1} = m-1$

I have no idea. Please help! $Z$ is used to represent the set of integers.

Well there exists only one multiplicative inverse to any element of $\mathbb{Z}_m$. So let's find out if the identity works:
$(m - 1)^{-1}(m - 1) = (m - 1)(m - 1) = m^2 - 2m + 1 \equiv 1~\text{mod m}$

It looks like the original assumption is true.

-Dan
• Dec 2nd 2007, 05:16 AM
Joel24
Yes, that makes sense. Thanks a lot Dan.