cat 2022 Question 21

Question

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

Accepted Answer

A is the HCF of $3^k + 4^k + 5^k$ for different values of k.

For k = 1, value is 12

For k = 2, value is 50

For k = 3, value is 216

HCF is 2. Therefore, A = 2

$4^k + 3(4^k) + 4^{k+2} = 4^k(1 + 3) + 4^{k+2} = 4^{k+1} + 4^{k+2} = 4^{k+1}(1 + 4) = 5 \cdot 4^{k+1}$

HCF of the values is when k = 1, i.e. $5 \ast 16 = 80$

Therefore, B = 80

A + B = 82

Question 21.

Question Explanation