Congruences modulo 8 - Corrigé

Énoncé

On suppose que \(x \equiv 1 \ [8]\) et \(y \equiv 3 \ [8]\) .

Déterminer des entiers congrus modulo \(8\) à :

1.  \(x+y\)

2.  \(x-y\)

3.  \(x+4y\)

4.  \(-5x+3y\)

5.  \(x^2+2y\)

6.  \(x+y^3\)

7.  \(-4x+y^2+5\)  

Solution

1. \(x+y \equiv 1+3 \equiv 4 \ [8]\)

2. \(x-y \equiv 1-3 \equiv -2 \equiv 6 \ [8]\)

3. \(x+4y \equiv 1+4 \times 3 \equiv 13 \equiv 5 \ [8]\)

4. \(-5x+3y \equiv -5 \times 1+3 \times 3 \equiv 4 \ [8]\)

5. \(x^2+2y \equiv 1^2+2 \times 3 \equiv 7 \ [8]\)

6. \(x+y^3 \equiv 1+3^3 \equiv 28 \equiv 4 \ [8]\)

7. \(-4x+y^2+5 \equiv -4 \times 1+3^2+5 \equiv 10 \equiv 2 \ [8]\)