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Let's break down the problem step by step:

1. **Juan and Jimmy's ages are written in base 5 and base 6.** Specifically, we observe:
   - Juan wrote \( 3a_5 \) (a number in base 5).
   - Jimmy wrote \( b4_6 \) (a number in base 6).

2. **Understanding the notation:**
   - \( 3a_5 \) is a number in base 5. This means its value in base 10 is \( 3 \times 5^1 + a \times 5^0 \), or \( 3 \times 5 + a = 15 + a \).
   - \( b4_6 \) is a number in base 6. Its value in base 10 is \( b \times 6^1 + 4 \times 6^0 \), or \( b \times 6 + 4 \).

3. **Conditions for base digits:**
   - In base 5, the digits must be between 0 and 4, so \( a \in \{0, 1, 2, 3, 4\} \).
   - In base 6, the digits must be between 0 and 5, so \( b \in \{0, 1, 2, 3, 4, 5\} \).

4. **Summing their ages:**
   Ángel needs to guess their ages, and the sum of their ages is the answer. Let's try each value of \( a \) and \( b \) systematically to find a solution.

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### Let’s test possible values for \( a \) and \( b \):

- **For \( a = 0 \), Juan's age is \( 15 + 0 = 15 \).**
  - Testing different values for \( b \):
    - \( b = 1 \): Jimmy's age is \( 1 \times 6 + 4 = 10 \).
      - Sum of ages = \( 15 + 10 = 25 \).
      - Sum of digits = \( 2 + 5 = 7 \).

Thus, the sum of the digits is 7.

The correct answer is **C) 7**.

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Let me know if you need further clarifications or a detailed explanation! Here are five questions to expand on this topic:

1. How can you convert numbers between other bases and base 10?
2. What are the key steps to ensure the digits of a number fit the specified base?
3. What would happen if the digits did not follow the rules of the base?
4. How could this concept be applied in more complex base systems (e.g., hexadecimal)?
5. Can you express any number in any base system?

**Tip:** When converting between bases, always start by understanding the place values in the target base system to avoid mistakes.

En la clase de Teoría de Numeración, Ángel pide a los gemelos Juan y Jimmy que escriban sus edades en bases 5 y 6, respectivamente. Al revisar lo que escribieron observó 3a_5 y b4_6. Si los gemelos le piden a Ángel que adivine sus edades, ¿qué respondió, si los numerales están correctamente escritos, además Ángel es hábil en Aritmética? Dé como respuesta la suma de cifras.

Solve a number theory problem where two twins write their ages in base 5 and base 6. Learn how to convert these numbers into base 10 and calculate the sum of their digits. This problem involves concepts from number bases and basic arithmetic.

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