What is the smallest number which must be added to 8261955 so as to obtain a…

Concept: A number is divisible by 11 exactly when the difference between the sum of its digits at odd positions (counted from the right, starting at the units digit) and the sum of its digits at even positions is 0 or a multiple of 11.

Applying it to this question:

1. Write 8261955 and mark digit positions from the right: units(1) = 5, (2) = 5, (3) = 9, (4) = 1, (5) = 6, (6) = 2, (7) = 8.

2. Sum the digits at odd positions (1st, 3rd, 5th, 7th from the right): 5 + 9 + 6 + 8 = 28.

3. Sum the digits at even positions (2nd, 4th, 6th from the right): 5 + 1 + 2 = 8.

4. Difference = 28 - 8 = 20, which is not a multiple of 11.

5. The units digit sits at an odd position, so adding a number to the units digit increases the odd-position sum -- and the difference -- by that same amount.

6. The next multiple of 11 above 20 is 22, so the difference needs to grow by 2: add 2 to the number.

7. New number = 8261955 + 2 = 8261957. Recomputed sums: odd positions 7 + 9 + 6 + 8 = 30, even positions 5 + 1 + 2 = 8, difference = 30 - 8 = 22, a multiple of 11.

Cross-check: Dividing directly, 8261955 divided by 11 gives quotient 751086 with remainder 9 (11 times 751086 = 8261946, and 8261955 minus 8261946 = 9). To reach the next exact multiple, add 11 - 9 = 2, matching the digit-sum method. Indeed 8261957 divided by 11 = 751087 exactly.

The smallest number that must be added to 8261955 to make it divisible by 11 is 2.