Apologies for the delay with this quiz's summary. Here ya go.

Long division is sometimes called _long division with remainders_ and

is done repeatedly dividing the divisor into each digit of the

dividend combined with the remainder of the previous division step.

For our example, 11 divided into 4096, the steps are:

1. Divide 11 into the first digit, 4; the result (i.e. first

digit of the quotient) is 0 with remainder 4.

2. Divide 11 into 40 (i.e. the previous remainder with the next

digit of the dividend); result is 3 with remainder 7.

3. Divide 11 into 79 (i.e. previous remainder 7 with next digit

9); result is 7 with remainder 2.

4. Divide 11 into 26; result is 2 with remainder 4.

And so our quotient is 0372, usually written without leading zeros as

372, with a remainder of 4.

In Ruby, it is trivial to get the end result without all the

intermediate steps:

quotient, remainder = dividend.divmod(divisor)

But this skips all the visible work that I wanted to see with this

quiz. Basically, your code really has to go through all the

intermediate steps in order to display them. With that in mind, let's

look at the solution from _Ken Bloom_ whose solution, while not

perfect, nicely separates the division logic from the display.

Here is Ken's main loop:

while dividend >= divisor

Math.log10(dividend).ceil.downto(0) do |exp|

magnitude = 10 ** exp

trydiv, rest = dividend.divmod(magnitude)

if trydiv >= divisor

exps << exp

dividends[-1] = trydiv

quotient_digit, remainder = trydiv.divmod(divisor)

products << quotient_digit * divisor

quotient += quotient_digit * magnitude

dividend = (remainder * magnitude + rest)

dividends << remainder

break

end

end

end

# display output

[quotient, dividend]

First, the outer loop exits once the dividend (updated during the

loop) becomes less than the divisor. At that point, what dividend

remains is the final remainder, and is returned to the caller along

with the quotient, as can be seen in the final line of the function,

`[quotient, dividend]`. This nicely follows the same convention as

`divmod` mentioned above.

The next loop takes the base-10 logarithm of the dividend, rounded up.

For 4096, this is 4 (because 4096 is more than 10**3 but less than

10**4). Essentially, this gives us the number of digits in the

dividend and the upper limit of digits in the quotient, and with the

next line:

magnitude = 10 ** exp

calculates successive, decreasing orders of magnitude to break down

the dividend. For example, first time through the inner loop, `exp` is

4, and `magnitude` is 10,000. That is used in the next line:

trydiv, rest = dividend.divmod(magnitude)

Although, in this example, the first time through the loop, `trydiv`

will be zero and `rest` will be 4096, the second time through (when

`exp` is 3 and `magnitude` is 1,000), `trydiv` becomes 4 and `rest`

becomes 96. It should be seen and understood by this that 4096 can be

composed of 4 * 1000 + 96. What Ken is accomplishing by all this is to

pull of the most significant digit of the dividend each time through

the loop: first is 4, then comes 0, then 9 and finally will be 6.

The condition that follows, `if trydiv >= divisor`, checks to see

whether the current digit for the quotient (at the current magnitude)

is something other than zero. If so, that digit is determined in the

line:

quotient_digit, remainder = trydiv.divmod(divisor)

For our example, we won't hit this line until `exp` is 2, when will

make `trydiv` will be 40 and `rest` will be 96. At this point,

`trydiv` is greater than `divisor`, and so the followup division gives

us `quotient_digit` as 3 and `remainder` as 7. (Note: 3 * 11 + 7 == 40.)

The next important part of handling the division is updating the

dividend, done in this line:

dividend = (remainder * magnitude + rest)

Continuing with the example, when `exp` is 2, `dividend` becomes 796

(i.e., 7 * 100 + 96). We can check the work by noting that the

quotient digit, 3, times the divisor and current magnitude (100) is

3300, and 3300 + 796 is our original dividend, 4096.

So, what are the other lines in the loop? Basically, Ken keeps a

record of the exponents, remainders, and dividends calculated during

the process, so these can be used in the output section to display the

long division.

If you have time, take a good look at _Sebastian Hungerecker_'s

solution. Not only does it handle the long division output, but can

calulate the decimal (instead of a remainder), even repeating, and do

division in a specified number base. Pretty sweet.