CAT 2023 Slot 2 QA Q22 Progressions & Series

Question 22 of the Quantitative Ability (QA) section, CAT 2023 Slot 2, with CAT's official answer and a worked solution.

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

TITA: type in the answer. There are no options and no negative marking.

Answer: 967 Progressions & Series TITA

Solution

  1. \(a_n = 6n + 7\) gives numbers \(\equiv 1 \pmod 6\) (from 13); \(b_n = 7n + 8\) gives numbers \(\equiv 1 \pmod 7\) (from 15).
  2. Common terms are \(\equiv 1 \pmod{42}\): 43, 85, 127, ...
  3. Largest three-digit one: \(42 \times 23 + 1 = 967\) (next is 1009).

Same concept in other years

See every question on Common terms of two APs form an AP with difference = LCM.

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