11. (10 points) Find the 6-digit number if
23 + 82 = 106411,
32 + 42 = 151613,
46 + 34 = 408113,
then 71 + 26 = ________.

The given problem presents a pattern where two numbers are combined in a unique way to produce a six-digit number. To solve this, we need to understand the pattern behind the operation of combining the two numbers.

We observe:

- \( 23 + 82 = 106411 \)
- \( 32 + 42 = 151613 \)
- \( 46 + 34 = 408113 \)

### Analysis of the Pattern
Let's break down each resulting six-digit number based on the inputs.

1. **First part**: 
   - In \( 23 + 82 = 106411 \), the first two digits "10" seem to correspond to the sum \( 23 + 82 = 105 \), so it might represent the **first digit(s)** of the sum of the two numbers.
   
   - Similarly, for \( 32 + 42 = 151613 \), the sum is \( 32 + 42 = 74 \), and the first digits are "15". It could be possible that the sum is multiplied by some factor or added with a constant to give the first three digits.

   - In \( 46 + 34 = 408113 \), the sum \( 46 + 34 = 80 \), and we see "408" as the first three digits.

2. **Second part**:
   - The middle two digits of each result appear to be consistently connected to the two individual numbers.
     - In \( 23 + 82 = 106411 \), "41" might be derived from \( 23 \times 2 = 46 \) or some similar rule.
     - Similarly, "16" in the second case and "81" in the third case likely follow a similar transformation rule related to the original numbers.

3. **Last part**:
   - The last two digits appear unrelated to the sum and might be directly linked to the second number.

Given these clues, let's predict the six-digit number for \( 71 + 26 \).

### Prediction for \( 71 + 26 \):
- First part: \( 71 + 26 = 97 \), and following a pattern similar to "15", "408", we might guess the first few digits are **971**.
  
- Second part: Based on transformations seen earlier, the middle two digits could correspond to a modified form of \( 71 \), resulting in something like **61**.

- Last part: We could guess the last two digits based on the transformations from 26, giving us something like **13**.

Thus, the number formed could be:

\[
971613
\]

Therefore, \( 71 + 26 = 971613 \).

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Would you like further details on the breakdown of the pattern?

### Related Questions:
1. How would you determine a consistent rule for the middle two digits?
2. Can we generalize this pattern for any two numbers?
3. What happens if we use different two-digit numbers? Would the pattern hold?
4. How is this problem related to modular arithmetic or base conversions?
5. Could there be a connection to a cryptographic or number-theoretic function?

**Tip**: Always look for common properties or transformations when dealing with number patterns.

11. (10 points) Find the 6-digit number if 23 + 82 = 106411, 32 + 42 = 151613, 46 + 34 = 408113, then 71 + 26 = ________.

Explore the solution to finding a 6-digit number based on given number combinations, such as 23 + 82 = 106411 and 32 + 42 = 151613. Learn how to break down number patterns and apply transformations to predict results. Suitable for Grades 9-12.

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