CAT 2023 Slot 2 QA Q3 Number Systems

Question 3 of the Quantitative Ability (QA) section, CAT 2023 Slot 2, with CAT's official answer and a worked solution.

For any natural numbers \(m, n\), and \(k\), such that \(k\) divides both \(m+2 n\) and \(3 m+4 n, k\) must be a common divisor of

A
\(2 m\) and \(3 n\)
B
\(m\) and \(2 n\)
C
\(2 m\) and \(n\)
D
\(m\) and \(n\)

Answer: (B) m and 2 n Number Systems MCQ

Solution

If \(k\) divides two numbers, it divides any integer combination of them.
\((3m+4n) - 2(m+2n) = m\), so \(k \mid m\).
\(3(m+2n) - (3m+4n) = 2n\), so \(k \mid 2n\).
So \(k\) is a common divisor of \(m\) and \(2n\). Option D fails: \(m=2, n=1\) gives \(4\) and \(10\), with \(k=2\) not dividing \(n=1\).

Shortcut. Eliminate \(n\) and then \(m\) by combining the two expressions.