Question

The number of positive integers n in the range $12\le n\le 40$ such that the product $\left( n-1 \right)\left( n-2 \right)...2.1$ is not divisible by n is
(a) 5
(b) 7
(c) 13
(d) 14

Answer 1

Hint: We are given that n is not divisible by $\left( n-1 \right)\left( n-2 \right)...2.1$ . Therefore, n must be a prime number since prime numbers will have only themselves and 1 as their factors. We have to find the number of prime numbers between 12 and 40.

Complete step by step answer:
We have to find the number of positive integers n in the range $12\le n\le 40$ such that the product $\left( n-1 \right)\left( n-2 \right)...2.1$ is not divisible by n. Since n cannot be divisible by the product $\left( n-1 \right)\left( n-2 \right)...2.1$ , n must be a prime number because prime numbers will have only themselves and 1 as their factors.
Let us assume $n=3$ .
$\Rightarrow \left( n-1 \right)\left( n-2 \right)...2.1=\left( 3-1 \right)\left( 3-2 \right)=2.1$
Clearly, we can see that $2\times 1=2$ is not divisible by 3.
Let us consider another prime number 5.
$\Rightarrow \left( n-1 \right)\left( n-2 \right)...2.1=\left( 5-1 \right)\left( 5-2 \right)\left( 5-3 \right)\left( 5-4 \right)=4\times 3\times 2\times 1=24$
Clearly, we can see that 24 is not divisible by 5.
Therefore, n is a prime number. Now, we have to find the number of prime numbers between 12 and 40. We know that the prime numbers between 12 and 40 are 13, 17, 19, 23, 29, 31 and 37. Hence, the number of prime numbers between 12 and 40 is 7.

So, the correct answer is “Option b”.

Note: Students should not get confused with prime numbers and composite numbers. Composite numbers will have factors other than themselves and 1. They must carefully list down the prime numbers between the given ranges because if any one of them is missed out, it will lead to the wrong solution.