CAT 2022 Slot 1 QA Question 7: 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(4^k) + 4^{

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(4^k) + 4^{

Accepted Answer

For $A$, check the parity first.
$3^k$ is odd, $4^k$ is even, and $5^k$ is odd.
So $3^k + 4^k + 5^k$ is always even.
For $k = 1$, value = $3 + 4 + 5 = 12$
For $k = 2$, value = $9 + 16 + 25 = 50$
$\gcd(12,50) = 2$
Hence, no integer larger than $2$ can divide every such number, and $A = 2$.
$4^k + 3(4