CAT 2022 Slot 1 Paper

Question : 52 of 66

Marks: +1, -0

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

Validate

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

4^{k+1}

HCF of the values is when

k=1

, i.e.

5 × 16=80

Therefore,

\B=80

\A+ \B=82

Go to Question:

Prev Question

Next Question