Question

The g.c.d of the smallest prime number and the smallest composite number is${\text{a}}{\text{. 1}} \\ {\text{b}}{\text{. 2}} \\ {\text{c}}{\text{. 0}} \\ {\text{d}}{\text{. 4}} \\$

Hint – The GCD (Greatest Common Divisor) of two numbers is the largest positive integer number that divides both the numbers without leaving any remainder, so use this property to reach the answer.

As we know composite numbers are the whole number that can be made by multiplying other whole numbers.
For example 6 can be made by multiplying $2 \times 3$, so 6 is a composite number, but 5 cannot be made by multiplying other whole numbers ($1 \times 5$ would work but we said to use other whole numbers).
So from above condition 1, 2, 3 cannot be a composite number because they cannot be made by multiplying other whole numbers.
So the smallest composite number $= 4$
Now as we know that the prime numbers are the multiplication of 1 and itself.
So the prime numbers be $\left( {2,3,5,7..............} \right)$
So the smallest prime number is 2.
Now we have to find out the g.c.d (greatest common divisor) of the smallest prime number and the smallest composite number.
As we know that the GCD (Greatest Common Divisor) of two numbers is the largest positive integer number that divides both the numbers without leaving any remainder.
So the factors of the smallest composite number $4 = 1 \times 2 \times 2$.
And the factors of the smallest prime number 2 $= 1 \times 2$.
So, from above we clearly say that the largest positive integer number that divides both the numbers 4 and 2 without leaving any remainder is 2.
So, the g.c.d of the smallest prime number and the smallest composite number is 2.
Hence option (b) is correct.

Note – Whenever we face such types of questions the key concept we have to remember is that always recall the property of composite number, prime number and GCD respectively which is stated above then use this property first find out the smallest prime and composite number respectively, then find the value of GCD using the property of GCD which is the required answer.