CAT 2022 Question Paper | Quant Slot 1

CAT Questions

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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Video Explanation

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Explanatory Answer

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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The answer is '82'