CAT 2022 Slot 1 QA Q21 Number Systems

Question 21 of the Quantitative Ability (QA) section, CAT 2022 Slot 1, with CAT's official answer and a worked solution.

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

TITA: type in the answer. There are no options and no negative marking.

Answer: 82 Number Systems TITA

Solution

\(3^k + 4^k + 5^k\): \(k = 1\) gives 12, \(k = 2\) gives 50; \(\gcd(12, 50) = 2\).
The sum is odd + even + odd = even for every \(k\), so \(A = 2\).
\(4^k + 3 \cdot 4^k + 16 \cdot 4^k = 20 \cdot 4^k\); smallest value is \(80\) at \(k = 1\) and every later value is a multiple of 80, so \(B = 80\).
\(A + B = 82\).