Question

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

Answer 1

Correct Answer: 82

Given the provided information, we can calculate the values of A and B as follows: 

For the numbers of the form 3^k + 4^k + 5^k: 

For k = 1: A = HCF(3¹ + 4¹ + 5¹) = HCF(12) = 12 

For k = 2: A = HCF(3² + 4² + 5²) = HCF(50) = 2 

For k = 3: A = HCF(3³ + 4³ + 5³) = HCF(216) = 2 

The highest common factor (HCF) of the values of A is 2. 

For the numbers of the form 4^k + 3(4^k) + 4^(k+2): 

For k = 1: B = 4¹ + 3(4¹) + 4⁽¹⁺²⁾ = 80 

For k = 2: B = 4² + 3(4²) + 4⁽²⁺²⁾ = 136 

For k = 3: B = 4³ + 3(4³) + 4⁽³⁺²⁾ = 560 

The highest common factor (HCF) of the values of B is 80. 

Therefore, A = 2 and B = 80. 

Hence, a + B = 2 + 80 = 82.

Answer 2

Step 1: Define \( A \) as the HCF of \( 3^k + 4^k + 5^k \)

We evaluate the expression for different values of \( k \):

• For \( k = 1 \): \( 3^1 + 4^1 + 5^1 = 12 \)
• For \( k = 2 \): \( 3^2 + 4^2 + 5^2 = 9 + 16 + 25 = 50 \)
• For \( k = 3 \): \( 3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216 \)

Now compute the HCF of the values:

\[ \text{HCF}(12, 50, 216) = 2 \Rightarrow A = 2 \]

Step 2: Define Expression for \( B \)

We consider the form:

\[ 4^k + 3 \cdot 4^k + 4^{k+2} \] Factor this expression: \[ = 4^k(1 + 3) + 4^{k+2} = 4^k \cdot 4 + 4^{k+2} = 4^{k+1} + 4^{k+2} \] \[ = 4^{k+1}(1 + 4) = 5 \cdot 4^{k+1} \] For \( k = 1 \): \[ B = 5 \cdot 4^2 = 5 \cdot 16 = 80 \]

Step 3: Add Values of \( A \) and \( B \)

\[ A + B = 2 + 80 = \boxed{82} \]

Final Answer:

\[ \boxed{82} \]