Science:Math Exam Resources/Courses/MATH312/December 2005/Question 02/Solution 1

The claims is false.

Let $2\equiv -4mod6$ so $a=8$ and $b=20$.

Further, let $8\equiv 20mod6$ so $c=2$ and $d=-4$.

So $c\mid a$ and $d\mid b$. Then

$\frac{a}{c}=4$ and $\frac{b}{d}=-5$ and so

$\frac{a}{c}\equiv 4mod6$ and $\frac{b}{d}\equiv -5\equiv 1mod6$

These two values are not equivalent and thus this gives a counter example.