CAT 2023 Slot 2 QA Question 1: 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

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

Accepted Answer

We are given that $k$ divides $m + 2n$.

So, $k$ also divides twice this expression.
$2(m + 2n) = 2m + 4n$
It is also given that $k$ divides $3m + 4n$. Hence, $k$ must divide the difference of these two expressions.
$(3m + 4n) - (2m + 4n) = m$
$\Rightarrow k	ext{ divides }m$

Now $k$ divides both $

For any natural numbers mmm, nnn and kkk, such that kkk divides both m+2nm + 2nm+2n and 3m+4n3m + 4n3m+4n, kkk must be a common divisor of

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