Answer
$m|a-b$, $a-b=km$, $a=b+km$
Work Step by Step
We are given that
$$a\equiv b \text{ (mod }m), m\text{ positive integer}.$$
We have to prove that
$$a\text{ (mod }m)\equiv b \text{ (mod }m).$$
Because $a\equiv b \text{ (mod }m)$, according to the definition it means that
$$m| (a-b)$$
therefore there is an integer constant $k$ so that we have:
$$a-b=km$$
which can be written as
$$a=b+km.$$
Now taking modulo of each side we get
$$a\text{ (mod }m)\equiv (b+km) \text{ (mod }m)$$
which means
$$a\text{ (mod }m)\equiv b \text{ (mod }m).$$
