Select the correct answerHow many different positive integers exist between 106 and 107, the sum of whose digits is equal to 2?Options7658

To determine how many different positive integers exist between $106$ and $107$ such that the sum of their digits equals $2$, we can break down the problem systematically.

1. ### Break Down the Problem

The range of integers we are considering is from $106$ to $107$, meaning:

• We need to identify integers between $106$ and $107$.

Since $107$ is not included when counting integers, we only need to check the integer $106$.

2. ### Relevant Concepts

• The integer $106$ is represented in its digit format as:
  • $1$ (hundreds place)
  • $0$ (tens place)
  • $6$ (units place)

We will check if there are any integers where the sum of the digits equals $2$.

3. ### Analysis and Detail

• The only digits available in the question are $0$, $1$, and $2$, since we are looking at digits that sum up to $2$.
• The only possible combinations of digits to make a three-digit number that fits in the range:
  • The first digit can only be $1$ because we need a three-digit number.
  • Therefore, our eligible numbers must fit the form:
    • $1XY$ where $X + Y = 1$ (as $1+X+Y$ should equal $2$)

The possible digits $X$ and $Y$ must also satisfy the constraint $X \geq 0$ and $Y \geq 0$. The valid pairs of $(X,Y)$ that meet these constraints are:

• $X=1, Y=0$ which gives us the number $110$
• $X=0, Y=1$ which gives us the number $101$
• $X=0, Y=0$ is not valid, since we cannot have only $1$ in the hundreds place for a three-digit number.

However, both $110$ and $101$ are not within the desired range $106$ to $107$.

4. ### Verify and Summarize

Within the constraints discussed, there are no three-digit numbers that exist strictly between $106$ and $107$ where the sum of the digits equals $2$.

Final Answer

The correct answer is $0$. There are no positive integers between $106$ and $107$ whose digits sum to $2$.