There are 818,181 7-digit numbers divisible by 11, the smallest being 1,000,010 and the largest 9,999,990.
In each of them S_odd - S_even=0 mod 11, where S_odd represents the sum of the digits on odd numbered places in the number and S_even represents the sum of the digits on the even-numbered places.
Example:
for 1234563
S_odd=1+3+5+3=12=
S_even=2+4+6


Let's assign a number N to the set of all 7-digit numbers divisible by 11 in which
N=MAX(S_odd , S_even).
We shall call this set set #N , and denote by q(N) the number of its members.

Examples:

EX1. q(1)=3, since there are 3 numbers possesing S_odd = S_even=1, namely:
1100000,1001000,1000010.

EX2. Set #36 includes all the numbers having a pattern 9a9b9c9 (so that S_odd is 36 and a+b+c=3 mod 11); so q(36)=number of compositions of number 3 into 3 non-negative integers, triplets like 111,102....003,030,300 +number of compositions of number 14 into 3 non-negative integers, triplets like 059,068,,,167,158,149,...770,...950 +number of compositions of number 25 into 3 non-negative integers i.e. 799,889,898,979,988,997.


a. Which is the most numerous set?
b. List q(i) for 1=1 to 36.

(Inspired by KS's "Divisibility and Digit Sum",
revised and vetted by Charlie.)