CAT 2023 Slot 2 QA Question & Solution
AlgebraMedium
Question
For any natural numbers m, n, and k, such that k divides both $m+2n$ and $3m+4n$, k must be a common divisor of
Options
m and n
2m and 3n
m and 2n
2m and n
Solution
It is given that k divides m+2n and 3m+4n.
Since k divides (m+2n), it implies k will also divide 3(m+2n). Therefore, k divides 3m+6n.
Similarly, we know that k divides 3m+4n.
We know that if two numbers a, and b both are divisible by c, then their difference (a-b) is also divisible by c.
By the same logic, we can say that (3m+6n)-(3m+4n) is divisible by k. Hence, 2n is also divisible by k.
Now, (m+2n) is divisible by k, it implies 2(m+2n) =2m+4n is also divisible by k.
Hence, (3m+4n)-(2m+4n) = m is also divisible by k.
Therefore, m, and 2n are also divisible by k.
The correct option is C