CAT 2023 Slot 2 QA 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

Question

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

Accepted Answer

Simplify the two sequence formulae.

$a_{n} = 13 + 6(n - 1) = 6n + 7$
$b_{n} = 15 + 7(n - 1) = 7n + 8$

Every term of the first sequence is 1 more than a multiple of 6.
$a_n \equiv 1\pmod{6}$

Every term of the second sequence is 1 more than a multiple of 7.
$b_n \equiv 1\pmod{7}$

So, a common term

Let ana_nan​ and bnb_nbn​ be two sequences such that an=13+6(n−1)a_n = 13 + 6(n - 1)an​=13+6(n−1) and bn=15+7(n−1)b_n = 15 + 7(n - 1)bn​=15+7(n−1) for all natural numbers nnn. Then, the largest three digit integer that is common to both these sequences, 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