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

Based on the provided data, the values of A and B are calculated as follows:

For 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) among the calculated A values is 2.

For 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) among the calculated B values is 80.

Consequently, A = 2 and B = 80.

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