Question

The set of odd integers is closed under
A) Addition
B) Subtraction
C) Multiplication
D) Division
E) None of these

Answer 1

In the given question, we have been asked if the set of odd integers is closed under the basic arithmetic operators and if it is, then under which one is it. For solving the question, we must know what odd integers are. Then we must know what being ‘closed’ under an operator means. Then we need to apply the meaning of being ‘closed’ under to the odd integers and check for it.

Complete Step by Step Solution:
Odd integers are the integers which are not divisible by $2$, i.e., their unit’s digit is $1,3,5,7,\,{\text{ or 9}}$.
When divided by $2$, they leave a remainder of $1$. They are represented as $2m + 1$.
If a set to be ‘closed’ under an operator, then it means that when the operator is applied on two numbers of the given set, we get another number of that set.
And clearly, the odd integers are closed under multiplication only, because addition or subtraction of two odd numbers always gives an even number. While dividing two odd integers, we might not get an integer (for example, $\dfrac{{15}}{{25}} = 0.6$ is not an integer).

Hence, the correct option is C).

Note: When solving such questions, we remove the wrong options by the method of elimination by bringing some examples which contradict the given statement. Then when that method is no longer deemed to be applicable on the given question, we apply the usual textbook methods to find the answer.