Can you solve this brain teaser?

Question

You can find mathematical puzzles and teasers like this one at http://www.mindchallenger.com The answers are there and new puzzles are added at the beginning of each month.

I would like to see the method you use to solve this and determine the numbers for each letter.

EVE/DID = 0.TALKTALKTALK...

Answer 1

Repeating decimal with four numbers ==> divisible by 9999. So 9999 must be a multiple of DID. Only three-digit candidate is 303.

With 303 as the denominator, E = 1 or 2
1 is less likely due to the quotients beginning in .3
Experiment with 2V2. V >=4, and 242/303 = .7986..., which works. So
E = 2
V = 4
D = 3
I = 0
T = 7
A = 9
L = 8
K = 6

Answer 2

Given that EVE / DID = 0.TALKTALKTALK...

Looking first at the repeating decimal 0.TALKTALKTALK..., recall that

0.TALKTALKTALK... = TALK/ 9999 = TALK / (10,000-1)

The denominator is the difference of two squares, so it factors easily:

0.TALKTALKTALK... = TALK / [(100+1)(100-1)]
  = TALK / (101*99) = TALK / (101*11*3*3)

The denominator, 101*11*3*3, contains two possible palindromic numbers, 303 and 909, suggesting that

EVE / DID is either EVE / 303 or EVE / 909

If one divides a palindromic number, 636, say, by 909, the repeating fraction is 0.6996... This is typical. A few trials show that the results are usually characterized by doubled numerals in the middle and a last digit “K” that is the same as “E”. The few exceptions are not solutions.

If DID is 303, then EVE must be smaller that 303, and may not use the digits 0 or 3. This leaves 121, 141, 151, 161, 171, 181, 191, 212, 242, 252, 262, 272, 282, and 292. EVE cannot be divisible by 3, since DID is divisible by 3. This reduces the list to 121, 151, 161, 181, 191, 212, 242, 262, 272, and 292.

Most of the lower digits are taken by DID and the possible values of EVE, so we want a relatively large repeating fraction. Beginning with EVE = 292, which isn’t a solution, and working down, we soon find that 242 / 303 = 0.7986 7986 7986...., and that is the answer.

To check this, note that 242 * 11 * 3 = 7986, and 303 * 11 * 3 = 9999,
so ( 242 * 11 * 3 ) / ( 303 * 11 * 3 ) = 7986 / 9999 = 0.7986 7986 7986...

Answer 3

Since the answer is less than one, DID is larger than EVE and D > E.
Since the answer is an infinitely-repeating decimal, EVE is not divisible by DID, but leaves a remainder of 0.

Turning the problem around, we have:
EVE= DID (0.TALKTALKTALK....)
or:
0.TALKTALKTALK . . .
multiplied by
DID

which equals the sum of:
or the sum of:
D00 x (0.TALKTALK....)
I0 x (0.TALKTALK....)
D x (0.TALKTALK....)
or (shifting as appropriate, and padding with leading zeroes):
= D x ( TA.LKTALK....)
+ I x ( 0T.ALKTALK....)
+ D x ( 00.TALKTALK....)

Now, combining the above, and writing each decimal column separately (but ignoring carries for the moment), we have:
Tens column (10): D*T
Units column (1): D*A + I*T
(DECIMAL POINT): .
Tenths column (0.1): D*L + I*A + D*T
Hundredths column (0.01): D*K + I*L + D*A
Thousandths column (0.001): D*T + I*K + D*L
next column (0.0001): D*A + I*T + D*K

next column: D*L + I*A + D*T
next column: D*K + I*L + D*A
next column: D*T + I*K + D*L
next column: D*A + I*T + D*K
and so forth.

The above does not include carryover, from column to column.

But the digit in the hundreds column must be E, which is equal to the carry from the tens column, which is D*T (plus a possible carry from the ones column. Likewise, the tens column is I, which must equal the ones digit of D* T plus a possible carry. Finally, the ones column is E, which must equal D*A + I*T plus a carry from the tenths column.

However, all of the remaining columns must be zero.
Therefore there must be a zero in the ones column from each of the following (plus possible carry from below):
D*L + I*A + D*T + (carry from the next lower column)
D*K + I*L + D*A + (carry from the next lower column)
D*T + I*K + D*L + (carry from the next lower column)
D*A + I*T + D*K + (carry from the next lower column)

That's as as far as I've gotten.
I'll continue later.

P.S. It might be possible to generate the the 90 possible values each for DID and EVE, multiply them out, and look for 4-digit repeats (where the 4 digits are not D, I, E, or V).
I might write a simple program to do that.
(But that would be "cheating", so I won't publish the answer.)

Answer 4

E < D and the result is a repeating decimal

Heck if I know