Why does $m$ dividing $a − b$ mean that $a$ and $b$ have the same remainer under division by $m$?
By the definition of a “remainder,” we can write $a = im + r_1$,
where $r_1$ is the remainder under division by m and satisfies $0≤r_1 ≤m−1$.
Similarly,
$b=jm+r_2$ with $0≤r_2 ≤m−1$. Then if $m$ divides $a − b$, this means that $m$ divides
$im + r_1 − jm − r_2 = m(i − j) + r_1 − r_2$.
Im stuck here !
Continuing from where you left off, this means that there is some $t$ such that: $m(i-j) + r_1 - r_2 = mt$, so $r_1 - r_2 = m(t-i+j)$, hence $m$ divides $r_1-r_2$. But $0 \le r_1 < m$ and $0 \le r_2 < m$, so $|r_1 - r_2| < m$. This can only mean that $r_1 - r_2 = 0$.