Let $$ A $$ be the largest positive integer that divides all the numbers of the form $$ 3^k + 4^k + 5^k $$, and $$ B $$ be the largest positive integer that divides all the numbers of the form $$ 4^k + 3\left(4^k\right) + 4^{k+2} $$, where $$ k $$ is any positive integer. Then $$ (A + B) $$ equals:

Question

Accepted Answer

To solve this problem:

1. **Finding $$ A $$**:  
   - The expression is $$ 3^k + 4^k + 5^k $$.  
   - Analyze divisibility for all values of $$ k $$.  
   - Clearly, $$ 2 $$ divides all terms because:  
     - $$ 3^k $$ is odd, $$ 4^k $$ is even, and $$ 5^k $$ is odd.  
     - The sum of odd + even + odd is always even.

- Therefore, the highest positive integer dividing all numbers of this form is:  
     $$
     A = 2.
     $$

2. **Finding $$ B $$**:  
   - The expression is $$ 4^k + 3\left(4^kight) + 4^{k+2} $$.  
   - Factorize and simplify:  
     $$
     B = 	ext{HCF of terms divisible by } \left(20 	imes 4^kight).
     $$
   - For any value of $$ k $$, the expression simplifies to:  
     $$
     B = 80.
     $$

3. **Sum of Divisors**:  
   Combine results:  
   $$
   A + B = 2 + 80 = 82.
   $$

Thus, the answer is **82**.