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The problem in the image states:

**"Al dividir tanto \(13^{ab}\) como \(17^{5a}\) entre 11, se obtiene 5 de residuo; ¿Cuál es el residuo de dividir \(13^{5b}\) entre \(a + b\)?"**

### Solution:

To solve this, we need to understand and utilize the given information effectively.

1. **Given Information:**
   - \(13^{ab} \equiv 5 \pmod{11}\)
   - \(17^{5a} \equiv 5 \pmod{11}\)

2. **Objective:**
   - Find the residue when \(13^{5b}\) is divided by \(a + b\).

### Step 1: Simplify using the modulo property
Since \(13^{ab} \equiv 5 \pmod{11}\), we know that:
\[ 13^{ab} \equiv 13^{k} \equiv 5 \pmod{11} \]
Where \(k = ab\).

### Step 2: Check the modular equivalences
We need to check the equivalence \(13^{k} \mod 11\). We can use Fermat's Little Theorem to simplify the calculations, where \(13 \equiv 2 \pmod{11}\). Thus, the problem becomes:
\[ 2^{ab} \equiv 5 \pmod{11} \]

Let's calculate the powers of \(2\) modulo \(11\):
\[ 2^1 = 2 \]
\[ 2^2 = 4 \]
\[ 2^3 = 8 \]
\[ 2^4 = 16 \equiv 5 \pmod{11} \]
\[ 2^5 = 32 \equiv 10 \pmod{11} \]
\[ 2^6 = 64 \equiv 9 \pmod{11} \]
\[ 2^7 = 128 \equiv 7 \pmod{11} \]
\[ 2^8 = 256 \equiv 3 \pmod{11} \]
\[ 2^9 = 512 \equiv 6 \pmod{11} \]
\[ 2^{10} = 1024 \equiv 1 \pmod{11} \]

So, \(2^{ab} \equiv 5 \pmod{11}\) implies \(ab\) must be congruent to \(4 \pmod{10}\).

### Step 3: Apply similar logic to the second congruence
Given that \(17^{5a} \equiv 5 \pmod{11}\), note that \(17 \equiv 6 \pmod{11}\). Therefore:
\[ 6^{5a} \equiv 5 \pmod{11} \]

Similarly, calculating powers of \(6\) modulo \(11\):
\[ 6^1 = 6 \]
\[ 6^2 = 36 \equiv 3 \pmod{11} \]
\[ 6^3 = 18 \equiv 7 \pmod{11} \]
\[ 6^4 = 42 \equiv 9 \pmod{11} \]
\[ 6^5 = 54 \equiv 10 \pmod{11} \]
\[ 6^6 = 60 \equiv 5 \pmod{11} \]

This implies \(5a\) must be congruent to \(5 \pmod{10}\). Therefore, \(a\) is congruent to \(1 \pmod{2}\).

### Step 4: Determine the residue of \(13^{5b} \mod (a + b)\)
Given \(a\) and \(b\) values:
- Since \(ab \equiv 4 \pmod{10}\) and \(a \equiv 1 \pmod{2}\), \(b\) must satisfy these equations. 
- \(13^{5b} \equiv 2^{5b} \pmod{11}\).

Use similar modular calculations to determine the residue when divided by \(a + b\).

### Final Step: Calculation and Conclusion
Without the exact value of \(a + b\), further simplification isn't possible without assuming specific \(a\) and \(b\) values satisfying the modular conditions.

Would you like to proceed with assuming specific values for \(a\) and \(b\), or do you have further details to specify these values?

Explore the solution to a complex problem involving modular arithmetic and Fermat's Little Theorem. Understand how to apply these concepts to find residues in modular equations. Suitable for advanced high school students and above.

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