**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.)