Chapter 4.3, Problem 8E

To determine

To state: If $s\equiv t\left(\mathrm{mod}k\right)$ is true.

Answer to Problem 8E

If $k|s$ and $k|t$ , $s\equiv t\left(\mathrm{mod}k\right)$ is true.

Explanation of Solution

Given:

  $k|s$ and $k|t$

  $a$ is congruent to $b\left(\mathrm{mod}n\right)$ if $n$ divides the difference $a-b$ .

  $s\equiv t\left(\mathrm{mod}k\right)$ can be written as $k|\left(s-t\right)$

Let $k=3$ , $s=15$ and $t=21$ .

Then, $k|s=3|15$ and $k|t=3|21$ is true.

  $k|\left(s-t\right)=3|15-21=3|-6$ is also true.