Question

The number of prime numbers among the numbers $105! + 2,105! + 3,105! + 4...105! + 4$and $105! + 105$ is
A) $31$
B) $32$
C) $33$
D) None of these

Answer 1

Hint: We should first know the definition of a prime number. The number which is only divisible by itself and $1$ are called prime numbers. We can say that they have only two factors. So we will first check in the above given factorial numbers if they have any common factors or not.
We will break the factorial number, as for example $5!$ can be written as
$5! = 5 \times 4 \times 3 \times 2 \times 1$

Complete step by step answer:
Here we have been given that
$105! + 2,105! + 3,105! + 4...105! + 4$and $105! + 105$
Let us take the first term i.e.
$105! + 2$
Here we can break the factorial term:
 $105! + 2 = 1 \times 2 \times 3 \times 4... \times 105 + 2$
By taking the common factor $2$ out, the above expression can also be written as $2(1 + 1 \times 3 \times 4 + ... \times 105)$
We can see that the above expression has a common factor, i.e. divisible by another number. So it is not a prime number.
Similarly we can take the third term:
 $105! + 3$
Here we can break the factorial term:
$105! + 3 = 1 \times 2 \times 3 \times 4... \times 105 + 3$
By taking the common factor $3$ out, the above expression can also be written as $3(1 + 1 \times 2 \times 4 + ... \times 105)$
Here also the expression has factors. So it is not a prime number.
Let us take another term i.e.
$105! + 101$
Here we can break the factorial term:
$105! + 101 = 1 \times 2 \times 3 \times 4... \times 101 \times 102 \times 103 \times 104 \times 105 + 101$
Here we can take the common factor $101$ out, the above expression can also be written as
$101(1 + 1 \times 2 \times 4 + ...100 \times 102 \times 103 \times 104 \times 105)$
So we can see that all the numbers from this set can be expressed in this form.
Therefore every number in the given set is divisible by the number added to the factorial, hence these are not prime.
Hence, the correct option is (D) None of these.

Note:
We should know that every Prime numbers can be expressed in the form as
$6n - 1$ Or, $6n + 1$ except the first five prime numbers where $n$ is a natural number. This form is not appropriate for the first two prime numbers.
Some properties of the prime numbers are:
Every number greater than $1$ can be divided by at least one prime number.
Every even positive integer greater than $2$ can be expressed as the sum of two primes.
Except $2,$ all other prime numbers are odd.