Question

Every ________ number can be expressed as a product of prime numbers.
(a) composite
(b) prime
(c) natural
(d) irrational

Answer 1

Hint: Here, we need to fill in the blank to form the correct statement using the given options. We will be using the definitions of the given options to find which of the given options is correct.

Complete step-by-step answer:
Prime numbers are the numbers which have only two factors, 1 and the number itself.
For example: 2 is the product of 2 and 1. Therefore, the factors of 2 are 2 and 1. Thus, 2 is a prime number.
Similarly, 5, 7, 11, etc. are prime numbers.
Now, let us take the prime number 11.
The prime number 11 can be expressed as the product of 1 and the prime number 11.
We know that 1 is neither a prime number not a composite number.
Thus, the prime number 11 cannot be expressed as a product of prime numbers.
Thus, option (b) is incorrect.
Composite numbers are the numbers which are not prime numbers. They are divisible by factors other than 1 and themselves.
For example: 4 is the product of 2 and 2. Therefore, 4 has a factor other than 1 and 4 itself. Thus, 4 is a composite number.
Now, let us take the composite number 60.
Since 60 is an even number, it is divisible by 2.
Therefore, we can write 60 as the product of 2 and 30.
Now, 30 is also an even number, that has the digit 0 at the unit’s place.
Therefore, 30 is also divisible by 2.
We can write 30 as the product of 15 and 2.
Thus, we can write 60 as the product of 2, 2, and 15.
Now, we know that 15 is the product of 3 and 5.
We can write 15 as the product of 3 and 5.
Thus, we can write 60 as the product of 2, 2, 3, and 5.
Here, 2, 3 and 5 are prime numbers.
Therefore, we can express 60 as the product of the prime numbers 2, 2, 3, and 5.
Similarly, every composite number can be expressed as a product of prime numbers.
Thus, the correct option is option (a).

Note: We will also check the other options.
A natural number is any integer that comes after 0.
For example, 1, 2, 3, 4, 143, etc. are all natural numbers.
Therefore, all prime and composite numbers are natural numbers.
Now, we have proved that prime numbers cannot be expressed as a product of prime numbers.
Thus, every natural number cannot be expressed as a product of prime numbers.
Therefore, option (c) is incorrect.
An irrational number is a number which is not a rational number. They cannot be written in the form \[\dfrac{p}{q}\]. For example, \[\sqrt 2 ,\sqrt 5 ,\sqrt 6 \], etc. are irrational numbers.
An irrational number cannot be written as a product of rational numbers.
Since all prime numbers are rational, every irrational number cannot be expressed as a product of prime numbers.
Therefore, option (d) is incorrect.