Unit Modulo n
is a unit modulo when some integer satisfies , equivalently when .
Definition 4.19 (Inverse and Unit Modulo )
Given , we say that is an inverse of modulo if .
We say that is invertible modulo , or that is a unit modulo , if such a exists.
Criterion
For , is a unit modulo if and only if :
In particular, when is prime every satisfies , so every non-zero residue is a unit modulo .
Cancellation
The inverse of a unit is unique: if , then
Hence is unambiguous, and cancellation by units follows: if is a unit and , then , by multiplying both sides by their inverses.
This is not true in general when the multiplier is not a unit: consider . Certainly .
Related
Stated in
- Definition 4.19 (Inverse and Unit Modulo )§4.5 Solving Congruences