If $j\mid (a-b)$ and $j\mid (b-c)$ then $j\mid (a-c)$ . Why is this true?

Question

Why is this true?

If $j\mid (a-b)$ and $j\mid (b-c)$ then $j\mid (a-c)$ .

Assuming all the variables stated are any integers.

Is it true because we use the $a\mid b$ is $ka=b$ definition?

Answer 1

$$j\mid b-c \implies b-c=k_1j ,k_1 \in \mathbb{Z}\\ j\mid a-b \implies a-b=k_2j ,k_2 \in \mathbb{Z}\\$$so by summation we have $$b-c=k_1j\\ a-b=k_2j \\a-c=(k_1+k_2)j \implies j\mid a-c \\\text{or} \\j\mid c-a$$