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CAT 2022 Slot 1 QA Question & Solution

Number SystemsHard

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 $4^k + 3(4^k) + 4^{k + 2}$ , where k is any positive integer. Then (A + B) equals

Solution

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

$4^k+3\left(4^k\right)+4^{k+2}=4^k\left(1+3\right)+4^{k+2}=4^{k+1}+4^{k+2}=4^{k+1}\left(1+4\right)=5\cdot4^{k+1}$

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

Therefore, B = 80

A + B = 82