11.1 The number 890 has the property that by changing any of its digits to 1 (increasing or decreasing), you can obtain a number that is a multiple of 11. Find the smallest three-digit number with the same property.
Answer: 120.

Solution. Since the last digit of the number must be changed to obtain a number divisible by 11, the required number should differ from it by 1. The smallest three-digit number divisible by 11 is 110. However, the numbers adjacent to it, 109 and 111, do not have the required property. Indeed, if the second digit of the number 109 is changed to 1, the only number that can be obtained is 119, which is not divisible by 11. If the first digit of the number 111 is changed to 1, the only number that can be obtained is 211, which is not divisible by 11.

The next three-digit number divisible by 11 is 121. Consider the number 120. Numbers that are multiples of 11 can be obtained from it according to the condition. The modified numbers 121, 110, 220 are divisible by 11.

Grading criteria.

«+» A complete and justified solution is provided.

«士» A justified solution with the answer 111 is provided, meaning the student considered that the first digit of the number can be zero (resulting in a two-digit number).

«Ғ» Only the correct answer (120 or 111) is provided, meaning it is not justified that this number is the smallest.

«-» The problem is not solved or is solved incorrectly.