Question

The set of all integral multiples of $5$ is a subgroup of

• A. The set of all rational numbers under multiplication.
• B. The set of all integers under multiplication.
• C. The set of all non zero rational numbers under multiplication.
• D. The set of all integers under addition.

Answer 1

Answer: (D)

The set of all integers under addition.

Let $H$ consist of the integral multiples of $5$, i.e.

$H=\{5x:x\in I\}$

This $H$ is the non empty subset of $I$.

Let $a=5r$ and $b=5s$ be any two elements of $H$, where $r$ and $s$ are some integers.

We have $a+b=5r+5s=5(r+s)\in H$.

$(r+s)$ comes out to be some integer.

$\therefore a\in H.,b\in H\Rightarrow (a+b)\in H$

Hence, $H$ is a subgroup of set of all integers under addition.