Let's solve this step-by-step.

The problem provides us with two different forms of expressions:

1. $A$ is the largest positive integer that divides all numbers of the form $3^k + 4^k + 5^k$.
2. $B$ is the largest positive integer that divides all numbers of the form $4^k + 3 \cdot 4^k + 4^{k+2}$.

We aim to determine the value of $A + B$.

Step 1: Finding $A$

To find $A$, we need to analyze the divisibility properties of the sequence defined by $3^k + 4^k + 5^k$.

Observing Patterns Modulo Small Primes

A good approach here is to consider the values of $3^k + 4^k + 5^k$ modulo small integers, such as modulo 3 and modulo 4.

1. Modulo 3:

   • $3^k \equiv 0 \pmod{3}$
   • $4^k \equiv 1^k \equiv 1 \pmod{3}$
   • $5^k \equiv 2^k \pmod{3}$

   So, $3^k + 4^k + 5^k \equiv 0 + 1 + 2 = 3 \equiv 0 \pmod{3}$.

   Therefore, $3$ divides $3^k + 4^k + 5^k$.

2. Modulo 4:

   • $3^k \equiv (-1)^k \pmod{4}$
   • $4^k \equiv 0 \pmod{4}$ (for $k \geq 1$)
   • $5^k \equiv 1^k \equiv 1 \pmod{4}$

   Depending on whether $k$ is even or odd:

   • If $k$ is even, $3^k \equiv 1 \pmod{4}$, so $3^k + 4^k + 5^k \equiv 1 + 0 + 1 = 2 \pmod{4}$.
   • If $k$ is odd, $3^k \equiv -1 \equiv 3 \pmod{4}$, so $3^k + 4^k + 5^k \equiv 3 + 0 + 1 = 4 \equiv 0 \pmod{4}$.

Thus, for $k$ odd, $4$ divides $3^k + 4^k + 5^k$. However, for $k$ even, only $2$ divides it.

Given this pattern, $A = 6$, as it is divisible by both $3$ and $2$.

Step 2: Finding $B$

For $B$, we analyze $4^k + 3 \cdot 4^k + 4^{k+2}$.

This expression simplifies as follows: $4^k + 3 \cdot 4^k + 4^{k+2} = 4^k(1 + 3 + 16) = 4^k \cdot 20.$

Thus, every term in this sequence is divisible by $20$, so the largest integer that divides all terms is $B = 20$.

Step 3: Calculating $A + B$

Now, $A = 6$ and $B = 20$, so: $A + B = 6 + 20 = 26.$

Final Answer:

$\boxed{26}$

Would you like further details or have any questions about this solution?

Here are five related questions to deepen understanding:

1. Why is it helpful to analyze sequences modulo small primes?
2. How can one determine the largest divisor for sequences involving powers?
3. What role does parity (odd/even) play in divisibility properties?
4. How would the solution change if the powers were different in each term?
5. Could we apply similar techniques to other types of exponential sequences?

Tip: When dealing with sequences of powers, modular arithmetic can reveal patterns that simplify finding common divisors.