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

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

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

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(4k) + 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.