CAT 2023 Question Paper | Quant Slot 2

CAT Questions

CAT 2023 Quant was dominated by Algebra followed by Arithmetic. 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 22 : Let \(a_n\) and \(b_n\) be two sequences such that \(a_n=13+6(n-1)\) and \(b_n=15+7(n-1)\) for all natural numbers \(n\). Then, the largest three digit integer that is common to both these sequences, is

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Video Explanation

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Explanatory Answer

\( a _ { n } = 13 + 6 ( n - 1 ) = 7 + 6 n \)
\( b _ { m } = 15 + 7 ( m - 1 ) = 8 + 7 m \)
For some m, n, let \( 7 + 6 n = 8 + 7 m \)
\( 6 n - 7 m = 1 \)
\( n = 6 , m = 5 \) works!
\( a _ { 6 } = 43 \) and \( b _ { 5 } = 43 \)
Since, \( L \cdot C \cdot M ( 6,7 ) = 42 \), the common terms of the series are of the form \( 43 + 42 k \)
Let’s find the smallest 4-digit number of this form…
\( \frac { 1000 } { 42 } \cong 23.80 \)
\( 43 + 42 ( 23 ) = 1009 \)
\( \therefore 43 + 42 ( 22 ) = 967 \) is the largest 3-digit such value.

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The question is " Let \(a_n\) and \(b_n\) be two sequences such that \(a_n=13+6(n-1)\) and \(b_n=15+7(n-1)\) for all natural numbers \(n\). Then, the largest three digit integer that is common to both these sequences, is "

The answer is '967'