Question

An integer $m$ is said to be related to another integer $n$ if $m$ is a multiple of $n$ . Then the relation is

• A. reflexive and symmetric
• B. reflexive and transitive
• C. symmetric and transitive
• D. equivalence relation.

Answer 1

The correct option is B reflexive and transitive

Given $mRn$ if $m$ is a multiple of $n,n,m\in I$
i.e $mRn$ if $n=km$ where $m,n,k$ are integers.
Now since, the multiple of every number is number also .
i.e. $n=1\times n\forall n\in I$
Hence, $R$ is reflexive.
$R$ is not symmetric as $6$ is a multiple of $2$ but $2$ is not a multiple of $6$.
Now, transitivity,
Let $mRn$ , $nRp$
$\Rightarrow m=k_{1}n$ and $n=k_{2}p$ where $k_1,k_2,,p\in I$
$\Rightarrow m=k_1{k}_{2}p$
$\Rightarrow m=kp$ where $k=k_1{k}_2$
Hence, $m$ is a multiple of $p$
Hence, $R$ is transitive.