# CAT 2022 Question Paper | Quant Slot 1

###### CAT Previous Year Paper | CAT Quant Questions | Question 21

CAT 2022 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 21 : 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

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 k $3 ^ { k } + 4 ^ { k } + 5 ^ { k }$ 1 12 2 50
Clearly 2 is the HCF of 12 and 50.
That does not mean 2 always divides $3 ^ { k } + 4 ^ { k } + 5 ^ { k }$.
We need to make sure that $3 ^ { k } + 4 ^ { k } + 5 ^ { k }$ is even for any value of k.
$3 ^ { k }$ is always Odd.
$4 ^ { k }$ is always Even.
$5 ^ { k }$ is always Odd.
(Odd) + (Even) + (Odd) = (Even)
Now, we can be sure that $3 ^ { k } + 4 ^ { k } + 5 ^ { k }$ is even for any value of k.
2 is the highest positive integer that divides all the numbers of the form $3 ^ { k } + 4 ^ { k } + 5 ^ { k }$.
A = 2
$4 ^ { k } + 3 ( 4 ^ { k } ) + 4 ^ { k + 2 } = 4 ^ { k } ( 1 + 3 + 16 ) = 20 $times 4 ^ { k }$ At k = 1, the value of $20 $times 4 ^ { k }$is 80. $4 ^ { k } + 3$ 4 ^ { k }$ + 4 ^ { k + 2 } = 20 $times 4 ^ { k } = 80 $times 4 ^ { k - 1 }$ For any other k, clearly $4 ^ { k } + 3$ 4 ^ { k }$ + 4 ^ { k + 2 }$ is divisible by 80.
80 is the highest positive integer that divides all the numbers of the form $4 ^ { k } + 3 ( 4 ^ { k } ) + 4 ^ { k + 2 }$.
B = 80
A + B = 82

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