Question

The solutions of $8x\equiv 6\left(mod14\right)$ are

• A. $\left[8\right],\left[6\right]$
• B. $\left[8\right],\left[14\right]$
• C. $\left[6\right],\left[13\right]$
• D. $\left[8\right],\left[14\right],\left[6\right]$

Answer 1

The correct option is C $\left[6\right],\left[13\right]$

$8x=6$ (mod 14)

$\Rightarrow \frac{8x-6}{14}=\lambda$

$\Rightarrow x=\frac{14\lambda +6}{8}$

$\Rightarrow x=\frac{7\lambda +3}{4}$

$=\frac{4\lambda +3\lambda +3}{4}$

$=\frac{4\lambda +3(\lambda +1)}{4}$

$=\lambda +\frac{3}{4}(\lambda +1)$

$\lambda ϵI^{+}$

8 and 14

GCD $=2$

when

$\lambda =2$ $x=3+\frac{3}{4}\times 4=6$

$\lambda =7$ $x=7+\frac{3}{4}\times 8=13$

$x=6,13$

Hence option C is correct