How many different positive integers exist between 106 and 107, the sum of whose digits is equal to 2?

Break Down the Problem

1. Identify the range of positive integers we are considering: between 106 and 107, which means we are looking at the numbers 107.
2. Determine the conditions for the digits of the numbers: the sum of the digits must equal 2.

Relevant Concepts

1. The integer range is strictly between 106 and 107, which implies we will only consider whole numbers starting from 107 (since 107 is the lowest integer greater than 106).
2. A three-digit number $abc$ must satisfy the equation: $a + b + c = 2$

Analysis and Detail

1. The number must be a three-digit number since we are looking at integers greater than 106.
2. The only possible three-digit number in our range is 107 itself.
3. Check the sum of the digits of 107:
   • Digits: $1, 0, 7$
   • Sum: $1 + 0 + 7 = 8$
4. The digit sum of 107 is 8, which does not satisfy our condition of the digit sum being equal to 2.

Verify and Summarize

• We have verified that the only number we considered in the range (107) does not meet the criteria since its digits do not sum to 2.
• Given that there are no other integers between 106 and 107, we conclude that no such integers exist.

Final Answer

There are 0 different positive integers between 106 and 107 whose digits sum to 2.