Question

Both the end digits of a 999 digit number N is 3 and N is divisible by 11, then find all the middle digits.

Answer 1

pattern observation: are the following numbers divisible by 3 ?
363 , yes
3113, 3223, 3333, ... ,3993, yes
35013, 34573..... , yes

For any natural number N, if you add up the odd-placed digits and if
you add up the even-placed digits, and you get the same result, then
N is divisible by 11.
e.g. for 35013
1st + 3rd + 5th digits = 3+0+3 = 6
2nd + 4th = 5 + 1 = 6

for 34573
3+5+3 = 11
4+7 = 11 , yes

try a larger number, as long as the sums as described above are equal
then the number is divisible by 11

See if you can make use of that, at the moment I am drawing a blank.
Seems like an onerous task if you want N to have 999 digits

Answer 2

just noticed a typo in my opening sentence.
Should have been:

pattern observation: are the following numbers divisible by 11 ?

Answer 3

Is answer 4???