### 11.8.11 Inverse in ℤ/pℤ

The inv or inverse command finds the inverse of an integer in ℤ/pℤ.

Since ℤ/pℤ is only a field if p is prime, the inverse is only guaranteed to exist if p is prime (and the integer is non-zero).

- inv takes n%p, an element of ℤ/pℤ.
- inv(n%p) returns the reciprocal of n%p in ℤ/pℤ.

#### Example

inv(3%13)

| ⎛ ⎝ | −4 | ⎞ ⎠ | %13

Indeed, 3·(−4)=−12≡ 1(mod 13 ).

You can also find the reciprocal using division:

1/(3%13)

| ⎛ ⎝ | −4 | ⎞ ⎠ | %13