Math 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

EVE/DID = 0.TALKTALKTALK...(1)
1000eve/did=talk.talktalk.....(2)
(2)-(1)
9999eve/did=talk
we could see here, did is a factor of 9999

9999=3.3.11.101
so DID is either 101, 303, or 909
but 101 cant do coz eve
moving on to talk
talk=9999eve/did
=11*eve or
talk=33*eve

if talk=11*eve, k=e, so no
talk=33*eve, making
9999/did=33, and
DID=303

eve is 3 digit no, so we know
101 E is obviously 2
talk=33*(202 + 10v)
=6666+330v
V cannot be 1, or t=6=k
possible V/A/L=4,5,7,8,9, and 1(except V)
possible T=7,8,9

10m+L=3V+6
10n+A=3V+6+m
T=6+n
so m, n cant be 0, and
A=11m+L-10n
=L+m (if m=n) or
A=L+m-10 (if m+1=n)

from T=6+n, 1<=n<=3,
so 1<=m<=2
because the most n can get is 1 more than m
now, m being either 1 or 2, restrict V to the set (4,5,7), (by 10m+L=3V+6)

and seeing that V's maximum possible,7, plus m's max possible,2, only reach 9(so no tens involve here);
we get A=L+m, and not A=L+m-10

now, n=m=1(if V=4) or 2(if V=5 or 7)
and this is the end of elimination round, and the start of trial and error.
from all possible V, only 4 will suffice an acceptable answer.
V=4, L=8, A=9, T=7

or maybe no TnE needed here.
possible A/L=4,5,7,8,9, and 1
possible T=7,8
possible V=4,5,7

if n=m=1, V=4, T=7
A=L+1 ; A and L are consecutive number
since 4 and 7 no longer available, that leave 8,9 for the taking

if n=m=2, T=8
A=L+2 ; A and L are either both odd or even
but V being either 5/7, they must be both odd, coz
3V+6=3(V+2)
==>odd plus even is odd, and
==>odd times odd is odd
(the above arrangement is interchangable)
if V=7, theres no possible match for A and L
if V=5, L,A could take 7,9 as answer.

here, now, we know A=9 either way, so from 10n+A=3V+6+m
9=3V+6-9n
3V=9n+3=3(3n+1)
V=3n+1
when n=1, V=4, and n=2, V=7

there, we got it finally
n=m=1, V=4, L=8, A=9, T=7

eve=242
did=303
talk=7986

Answer 2

EVE/DID = 0.TALKTALKTALK... = TALK/9999
Obviously there is common factor/s between TALK and 9999.
Let us try to factorize 9999:
the prime factorization of 9999 = 3^2*1111 = 3^2*11*101. Therefore, DID must be either 101, 303, or 909.
11 must be a common factor of 9999 and TALK, such that 33*EVE would be TALK. 11*EVE is impossible to be TALK, since the last digit of 1*EVE is still "E" not "K". 99*EVE would be TALK, either, since EVE can not be "101" (can you see why?) and is at least 121, hence 99*EVE would be otherwise of 5-digits. Therefore, DID is 303. Excluding digits 0 and 3 in both EVE and TALK, we list EVE, and 33*EVE as four possibilities: 212, 6996; 242, 7986; 262, 8646; and 272, 8976. Rejecting the number with repeated digit in 33*EVE, or any identical digit as in EVE, we get the only solution:
242/303 = 0.798679867986798679867986....