Which number among 11368, 11638, 11863 and 12638 is divisible by 11?

To determine which of the given numbers is divisible by 11, we can use the divisibility rule for 11:
A number is divisible by 11 if the difference between the sum of the digits at odd positions (1st, 3rd, 5th, etc.) and the sum of the digits at even positions (2nd, 4th, 6th, etc.) is either 0 or a multiple of 11.

Let's test the given options step-by-step:

1. Test the number 11638:
Let's identify the positions of the digits from left to right: 1 (1st), 1 (2nd), 6 (3rd), 3 (4th), and 8 (5th).
Sum of the digits at odd positions (1st, 3rd, 5th):
$1+6+8=15$
Sum of the digits at even positions (2nd, 4th):
$1+3=4$
Now, find the difference between the two sums:
$15-4=11$
Since the difference is 11 (which is divisible by 11), the number 11638 is divisible by 11.

2. Test the number 11368:
Sum of the digits at odd positions: $1+3+8=12$
Sum of the digits at even positions: $1+6=7$
Difference: $12-7=5$ (not divisible by 11).

3. Test the number 11863:
Sum of the digits at odd positions: $1+8+3=12$
Sum of the digits at even positions: $1+6=7$
Difference: $12-7=5$ (not divisible by 11).

4. Test the number 12638:
Sum of the digits at odd positions: $1+6+8=15$
Sum of the digits at even positions: $2+3=5$
Difference: $15-5=10$ (not divisible by 11).

Therefore, the only number divisible by 11 is 11638.