To solve this problem, let's consider a few important points about digits and numbers.
1. Since we can only use the digits 0 through 9 (which is a total of 10 different digits) to make an 11-digit number, and only one digit is repeated with no other repetitions, we can conclude that the repeated digit must be one of the digits from 0 to 9.
2. If we add up all the digits from 0 to 9, we get a total sum of:
0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45.
3. According to the problem, the sum of the digits of the 11-digit number is 49. This sum can be thought of as the sum of the unique digits (which we already know is 45) plus the repeated digit. In other words, the repeated digit must account for the excess of this total sum over the sum of the unique digits, which is 49 - 45.
Now, let's calculate the repeated digit:
49 (sum of the digits of the 11-digit number) - 45 (sum of the unique digits from 0 to 9) = 4.
The repeated digit, therefore, is 4.
