Could u please show me specific process of this question I r

Could u please show me specific process of this question?

I really have no idea how to do these questions

Thanks a lot!

1. Prove the observation that, n! +1 is not divisible by any prime less than n, and use it to prove that there is no largest prime number.

Answer: 1.a) Let us first prove that (n! + 1) is not divisible by any prime no. less than ‘n’.

Let us take some example first before proceed ahead.

Let n= 5,

So, (n! + 1) is = (5! + 1) = (5*4*3*2*1 + 1) = (120 + 1) = 121

The prime nos. less than n= 5 is 3 & 2.

So, the no. 121 is not divisible by 3. Also, the no. 121 is not divisible by 2.

Hence, it is true that the (n! + 1) is not divisible prime nos. less than n, here for n = 5.

Let n= 8,

So, (n! + 1) is = (8! + 1) = (8*7*6*5*4*3*2*1 + 1) = (40320 + 1) = 40321

The prime nos. less than n= 8 is 7, 5, 3 & 2.

So, the no. 40321 is not divisible by any of 7, 5, 3 & 2.

Hence, it is true that then (n! + 1) is not divisible by prime nos. less than n, here for n = 8.

Let us prove this by Induction Method.

Let us assume, it is true for n = k that (n!+ 1) is not divisible by prime nos. less than ‘n’.

any prime nos. less than ‘k’. (Assumption)

So, [(k+1)! + 1] = {[(k+1)*k*(k-1)*(k-2)*…..*1] + 1} is not divisible by any prime no. less than (k+1).

Proved.

1.b) Let us now prove that that there is no largest prime number.

Let here that n = 100,

Then (n! + 1) = 9.332621544 x 10^157 + 1 = which is big no.

But the prime no is less than 100.

Let again here that n = 1000,

Then (n! + 1) = 4.023872601 x 10^2567 + 1 = which is very big no.

But the prime no is less than 1000.

In this way, we say that the numbers are very big and endless but the prime no. lesser than that number. Hence, there is no largest prime numbers.

Thank You