Let's break down the pattern step-by-step to understand how to find the last two digits of the sequence.

Analyzing the Pattern:

We are given the following information:

1. $23 + 82 = 106411$
2. $32 + 42 = 151613$
3. $46 + 34 = 408113$

For each equation, the result is a 6-digit number, and we need to identify the sequence behind each pair of digits. Let's identify the sequence of each segment systematically.

Step 1: First Two Digits

The first two digits come from the sum of the digits in the first number, multiplied by the first digit:

• $23 + 82 = 106411 \Rightarrow (2 + 3) \times 2 = 5 \times 2 = 10$
• $32 + 42 = 151613 \Rightarrow (3 + 2) \times 3 = 5 \times 3 = 15$
• $46 + 34 = 408113 \Rightarrow (4 + 6) \times 4 = 10 \times 4 = 40$

Thus, for $71 + 26$:

• $(7 + 1) \times 7 = 8 \times 7 = 56$

First two digits: 56.

Step 2: Third and Fourth Digits

The next two digits are derived from raising the first digit of the second number to a certain power:

• $23 + 82 = 106411 \Rightarrow 8^2 = 64$
• $32 + 42 = 151613 \Rightarrow 4^2 = 16$
• $46 + 34 = 408113 \Rightarrow 4^3 = 64$

For $71 + 26$:

• $2^6 = 64$

Third and fourth digits: 64.

Step 3: Fifth and Sixth Digits

The last two digits in each equation follow a consistent pattern:

• $23 + 82 = 106411 \Rightarrow 11$
• $32 + 42 = 151613 \Rightarrow 13$
• $46 + 34 = 408113 \Rightarrow 13$

These numbers could be based on a cyclic pattern or some operation on the digits of the numbers. After analyzing, it appears that the pattern is arbitrary, repeating 11 and 13.

Final Answer:

Given that the fifth and sixth digits for $71 + 26$ should follow the pattern, we have:

$\boxed{566413}$

If you have any questions or want more details on any step, let me know!

Related Questions:

1. How would the solution change if we modify the second number in each equation?
2. Can we create a formula for the fifth and sixth digits based on previous patterns?
3. What if the first number changes significantly—will the pattern still hold?
4. Are there other operations (e.g., multiplication) that yield the same results?
5. Can we automate this process for any arbitrary set of numbers?

Tip:

Whenever tackling a pattern problem, break the solution into small segments, and analyze each separately before combining them into the final answer.