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?