Let’s analyze Tomas’s conjecture.

Tomas claims that when you multiply a two-digit number by 11, the first and last digits of the resulting product match the original two-digit number. Let’s test this by breaking down the examples provided and using additional ones for verification.

Examples from the problem:

1. $17 \times 11 = 187$

   • The original number is 17.
   • The first digit of 187 is 1, and the last digit is 7. This matches 17.

2. $23 \times 11 = 253$

   • The original number is 23.
   • The first digit of 253 is 2, and the last digit is 3. This matches 23.

3. $41 \times 11 = 451$

   • The original number is 41.
   • The first digit of 451 is 4, and the last digit is 1. This matches 41.

4. $62 \times 11 = 682$

   • The original number is 62.
   • The first digit of 682 is 6, and the last digit is 2. This matches 62.

Explanation of why this works:

When a two-digit number $AB$ (where $A$ is the tens digit and $B$ is the units digit) is multiplied by 11, the product can be computed as: $AB \times 11 = (10A + B) \times 11 = 110A + 11B = 100A + (A + B) \times 10 + B$ This creates a three-digit number $A \, (A + B) \, B$ unless $A + B$ exceeds 9 (causing a carry).

Tomas's conjecture is reasonable as long as there’s no carry when $A + B > 9$. To disprove the conjecture, consider $59 \times 11$:

$59 \times 11 = 649$ Here, the original number is 59, but the product's first and last digits are 6 and 9, not 5 and 9. This counterexample shows the conjecture only holds when $A + B \leq 9$.

Would you like more detailed steps or further examples?

Relative Questions:

1. What happens when the sum of the digits $A + B > 9$?
2. Can this pattern be generalized to three-digit numbers multiplied by 11?
3. Why does carrying affect the result of Tomas's conjecture?
4. How does the distributive property help explain this phenomenon?
5. What mathematical proof could solidify the limitations of Tomas’s conjecture?

Tip: Test conjectures with counterexamples to confirm their validity across all cases.