To solve the problem $AB \times AB = ABB$, where $AB$ is a two-digit number and $ABB$ is a three-digit number, and each letter represents a different digit:

Step-by-Step Solution:

1. Let $AB = 10A + B$:

   • $AB$ is a two-digit number where $A$ and $B$ are digits. So, $AB = 10A + B$.

2. Equation Setup:

   • $AB \times AB = ABB$ translates to: $(10A + B) \times (10A + B) = 100A + 10B + B$ Simplify the right side: $(10A + B)^2 = 100A + 11B$

3. Range Constraints:

   • $AB$ is a two-digit number ($10 \leq AB \leq 99$).
   • $ABB$ is a three-digit number ($100 \leq ABB \leq 999$).
   • Therefore, $10A + B$ should satisfy $100 \leq (10A + B)^2 \leq 999$.

4. Trial and Error with Valid $A$ and $B$:

   • Start with reasonable $AB$ values and calculate $(10A + B)^2$ to see if it equals $ABB$.

Calculation:

• $A = 2$, $B = 5$: $AB = 25, \quad AB \times AB = 25 \times 25 = 625, \quad ABB = 625$ This satisfies the equation.

Solution:

The decoded numbers are: $A = 2, \quad B = 5, \quad AB = 25, \quad ABB = 625.$

Let me know if you’d like further clarifications or additional examples.

5 Related Questions:

1. What are the steps to verify if another pair of digits satisfies the equation $AB \times AB = ABB$?
2. Can the problem be solved using a systematic algebraic approach instead of trial and error?
3. How many unique solutions exist for equations of this form where $AB$ is two digits?
4. How would you extend this problem to work for $ABC \times ABC = ABCC$?
5. What are other common cryptarithm puzzles involving multiplication?

Tip:

In cryptarithm puzzles, always identify constraints and logical limits for the numbers to narrow down possibilities efficiently.