Q:

casting out nines: division

the division in the casting out nines is not clear to me. In the given example he got a 8 and a 4, and later he got an 8 and a 3. Shouldn't it be an 8 and a 4 (or 3) in both?

A:

The example seems to be:  
$$275462 \div 877 = 314 \quad \text{ r. } \quad 84 $$ which which can be checked by casting out nines and verifying  $$\quad 8 \quad \quad \equiv \quad 4 \quad\times \quad 8 \: \quad + \: \quad 3$$ (modulo 9).
In fact what is being checked is the equivalent multiplication statement
$$275462 = 877 \times 314 \quad + \quad 84 $$
which could also be written as 
$$(30606 \times 9 + 8) = (97 \times 9 + 4)\times(34\times 9 +8) +(9 \times 9 + 3)$$
and (modulo 9) all the $\times 9$ terms are equivalent to 0.

