Question

An integer ${\mu }_1$ is said to be related to another integer ${\mu }_2$ if ${\mu }_1$ is multiple of ${\mu }_2$. This relation is

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

Answer 1

The correct option is D reflexive and transitive

We know that a number is a factor as well as multiple of itself.

Since 'n' will be a factor of 'n' for any integer, hence $(n,n)$ will be a subset of the relation. In other words $nRn$ for a number 'n' holds true. This makes the relation reflexive.

If 'n' is a factor of 'm' then simultaneously 'm' cannot be a factor of 'n' since $n\le m$.

Thus $mRn\to nRm$ is not true. Thus the relation is not symmetric.

Now, if 'm' is multiple of 'n' and 'p' be a multiple of 'm', (all being distinct integers), then 'n' will be a common factor of 'm' and 'p'. Thus

$pRm\to mRn\to pRn$ is true.

Hence the relation is transitive.