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?

• why the down vote? – Rudy the Reindeer Apr 4 '11 at 12:54
• No. $8$ and $4$ are the residues for the dividend and divisor, $8$ and $3$ are the residues for the quotient and the remainder. There's no reason for these to be the same. The only relation to be expected among them is the one given in the check, namely $8=4\cdot8+3\pmod{9}$. – joriki Apr 4 '11 at 12:59
• It would be much more helpful to reproduce the problem here so people don't have to click through. Then please add some more detail in the question showing what you don't understand which would make it easier to answer. – Ross Millikan Apr 4 '11 at 13:01
• It's using $\rm\displaystyle\ \frac{a}b\ =\ c + \frac{d}b\ \Rightarrow\ a\ \equiv\ b\ c + d\ (mod\ 9)$ – Bill Dubuque Apr 4 '11 at 13:25
• The tag "division-algebra" ought to mean something different. – Douglas Zare Apr 4 '11 at 19:41

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.

• Ow, I got it, the 4x8+3 is for checking the division.. I think I was too asleep to other day.. XD Thanks! – Tom Brito Apr 10 '11 at 12:46