Think Carefully Part 2

There are infinitely many primes, and there does not exist the largest prime, and so there does not exist last digit of the largest prime number.

See this if you want to know the proof of the existence of infinitely many primes. For any clarifications on this, comment below.

NOTE :- The largest known prime till date is $2^{57885161} - 1$ , whose last digit can be calculated by basic modular arithmetic :- $2^{57885161} \equiv 2$ ( mod 10 ) $\Rightarrow 2^{57885161}-1 \equiv 1$ ( mod 10 ), and hence the last digit of the largest known prime is $\boxed{1}$ . For anyone interested in knowing about modular arithmetic, check out the Brilliant Wikis on it here . Have Fun !

There are infinitely many prime numbers, so there is no 'largest prime'.

Proof:

Assume there are finitely many prime numbers, $p_1 , p_2 , p_3 , ... , p_n$

Let

$A= p_1 p_2 p_3 ... p_n +1$

Then none of $p_1 , p_2 , p_3 , ... , p_n$ can divide A ,

we consider 2 cases, (i) A is prime and (ii) A is not prime.

If A is prime, we got (n+1)th prime, which is A and A is not in the list of n primes

If A is not prime, A can be factorized into multiple of few primes, but none of $p_1 , p_2 , p_3 , ... , p_n$ can divide A, so there must be new primes, which can divide A.

And once we got a new list of prime numbers, we can repeat it many times and hence there are infinitely many prime numbers.