Question

Solve the congruence $2\mathrm{x}=7\left(\mathrm{mod}17\right)$using the inverse of 2 modulo 7 found in part (a) of Exercise 6.

Short answer

The congruence is solved

$2\mathrm{x}=12\left(\mathrm{mod}17\right)$

Step-by-step solution

Step 1

We have to solve this

$2\mathrm{x}=7\left(\mathrm{mod}17\right)$

We use Bezout’s Theorem to express $\gcd \left(17,2\right)$, as a lineal combination of 17 and 2. Thus we can observe that the inverse of 17is 2, but if we add 17 to -8, we obtain 9, so the inverse is, as well, 9

$17=(2\ast 8)+1$

So

$1=(17\ast 1)-(2\ast 8)=(17\ast 1)+\left(\right(-8)\ast 2)$

Step 2

So we can multiply both sides by 9 to obtain that x is equivalent with $12\mathrm{mod}17$

$$\begin{gathered} 9\ast 2x=9\ast 7(mod17) \\ x\equiv 63(mod17) \end{gathered}$$

_{$x\equiv 12(mod17)$}