If x is a positive integer, is x! + (x + 1) a prime number?

Dear Radhika11

Here's an alternate solution.

We'll first analyze the question statement and only then go to St. 1 and 2.

Given: x is a positive integer. So, possible values of x: 1, 2, 3, 4 . . .

To find: If x! + x + 1 is Prime

Analysis:

For x = 1, x! = 1
And, x! + x +1 = 1 + 1 + 1 = 3, which is a Prime number

For x > 1,

x! will always be an even number (Because, x! contains the product of consecutive integers. So, this product will have Even AND odd terms. We know that when an even number is multiplied with anything, the product is always even)

So, the sum (x! + x + 1) = (An even number + x + Odd number) = (Odd number + X)

Even for the lowest possible value of x (x = 1), the value of the sum (x! + x + 1) was equal to 3.

So, for x > 1, the value of this sum is definitely going to be greater than 3.

And, we know that all Prime numbers greater than 2 are odd.

So, the sum (Odd number +X) can be prime only if first, this sum is an odd number.

That is, if, X is an even number.

So, for x > 1, x being an even number is a NECESSARY condition for the sum (x! + x + 1) to be prime.

But is it a SUFFICIENT condition? That is, can you say that if x is even, that must mean that the sum (x! + x + 1) will be prime?

Let's see:

x x! + x + 1

2 2! + 2 + 1 = 7 (Prime)
4 4! + 4 + 1 = 29 (Prime)
6 6! + 6 + 1 = 727 (Prime)
8 8! + 9 (both terms in this sum are divisible by 3) (NOT Prime)

Thus, we see that for some even values of x, the sum (x! + x + 1) will be Prime and for others, it will not be.

With this understanding, let's now look at the two statements:

Statement 1: x < 10

As per this statement, x can be even (=> some possibility of the sum (x! + x + 1) to be prime)
or x can be odd (=> NO possibility of the sum (x! + x + 1) to be prime)

Clearly not sufficient.

Statement 2: x is even

As illustrated in our analysis of the question statement, x being Even is not a sufficient condition for the sum to be prime.

Not Sufficient.

Statement 1 + 2

=> x is an even number < 10

As we saw in our analysis above, for x = 8, the sum is not prime. For all other possible values of x, the sum is prime.

Not Sufficient.

Therefore, correct answer: Option E

(Note: Had Statement 1 been: x < 8, then the correct answer would have been Option C)

I hope this helped!

Japinder