Q7. - Algebra 2

Question

Which of the following sets of numbers has the property that the product of any two numbers is also a number in the set.

I the set of positive numbers

II the set of prime numbers

III the set of even integers

 

$\begin{array}{l}\text{A}.\text{ I only B. II only}\\ \text{C}.\text{ IIII only D. I and III only}\end{array}$

Step-by-Step Solution

Answer

The set of positive numbers and set of even integers satisfy the property. Option D is correct.

Step-1 – Apply the concept of types of numbers

The set of positive numbers are those numbers which are greater than 0.

The set of prime numbers are those numbers which have only two factors that are 1 and number itself.

The set of even integers are those integers which are divisible by 2.

Step-2 –Apply the property to positive numbers

Consider the set of positive numbers.

Example of positive numbers are 1, 10, and many more.

Now, take any two arbitrary positive numbers say 2 and 5.

Next, the multiplication of 2 and 5 is computed below.

$2\times 5=10$

The obtained number is again a positive number.

Thus, the product of any two positive numbers is also a positive number.

Step-3 –Apply the property to prime numbers

Consider the set of prime numbers.

Example of prime numbers are 2, 5, 7, 11 and many more.

Now, take any two arbitrary prime numbers say 2 and 5.

Next, the multiplication of 2 and 5 is computed below.

$2\times 5=10$

The obtained number 10 is not a prime number. 10 is a composite number as it has factors 2, 5, 1 and 10.

Thus, the product of any two prime numbers is not a prime number.

Step-4 –Apply the property to even integers

Consider the set of even integers.

Example of even integers are 2, 4, 6 and many more.

Now, take any two arbitrary positive numbers say 2 and 4.

Next, the multiplication of 2 and 4 is computed below.

$2\times 4=8$

The obtained number is again an even integer.

Thus, the product of any two even integers is also an even integer.

Hence, the set of positive numbers and set of even integers satisfy the property.