Euclidean remainder: rem

### 5.34.6 Euclidean remainder: rem

The rem command finds the remainder of the Euclidean division of two polynomials (see also Section 5.28.3).

- rem takes two mandatory arguments and one optional argument:
  - P and Q, two polynomials with coefficients in ℤ/pℤ.
  - Optionally x, the variable (by default x), if P and Q are given as expressions.
- rem(P,Q ⟨,x⟩) returns the remainder of the Euclidean division of P divided by Q.

#### Example.

Input:

rem((x^3+x^2+1)%13,(2*x^2+4)%13)

Output:

⎛ ⎝ | ⎛ ⎝ | −2 | ⎞ ⎠ | %13 | ⎞ ⎠ | x+ | ⎛ ⎝ | −1 | ⎞ ⎠ | %13

Indeed x3+x2+1=(2x2+4)(x+1/2)+5x−4/4 and −3*4=−6*2=1 mod13.