Question

If $a$ and $b$ are two integers, then which of the following may not be an integer?

• A. $a+b$
• B. $a-b$
• C. $a\times b$
• D. $a÷b$

Answer 1

The correct option is D

$a÷b$

Explanation for the correct option:

A rational number is a number that can be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are co prime integers and $q\ne 0$.

An integer is a rational number where $q$ is a factor of $p$.

Thus, when $q$ is not a factor of $p$, $\frac{p}{q}$ results in a number that is not an integer.

So, in $a÷b$, when $b$ is not a factor of $a$, the resulting number will not be an integer.

For example: Consider the integers $a=10$ and $b=9$

Here,

$$\begin{gathered} a÷b=10÷9 \\ =1.111 \end{gathered}$$

Thus $a÷b$ is not an integer in this case as $b$ is not a factor of $a$

Consider the integers $a=10$ and $b=5$

Here,

$$\begin{gathered} a÷b=10÷5 \\ =2 \end{gathered}$$

Thus $a÷b$ is an integer in this case as $b$ is a factor of $a$

Hence, option D is the correct.