Hard work or guess work?

Because 2 is a prime number, this sum is odd.

there are 13986 prime numbers. out of which 2 is also included in the sum. so there are 13985 prime numbers left. and its sum cant be even.

so 2 + ( sum of odd 13985 prime numbers ) = 2 + something odd = odd

Hence it can not be perfectly divided by any even number.

you can write first 5 prime no. and see that if we take 2 prime no their sum is odd and if we take 3 prime no their sum is even .so its simple

look, its easy, there's no point thinking our asses off.

Think it this way

2, 3, 5, 7

2 + 3 = 5, odd number 2 + 3 + 5 = 10 even number 2 + 3 + 5 + 7 = 17 odd number

This game continues for ever

now odd number is not divisible by 2 and even number is.

For the divisibility of numbers by 4, 6, and 8, there are different approaches. We do not want to use it here. Just think simple, the answer should be None of this.

Solved in the first try ;)

Clearly 2 is included in this sum . This leaves us with 13985 prime numbers of which all are odd. Sum of these odd numbers will be odd and when 2 is added then the resultant sum will also be add.

Since no even number divides an odd number therefore none of these is the answer.

since 2 is only no., and other 13985 are odd so if we multiply 13895 odd no. with just one even no. 2, we will get a no. divisible by odd nos.

13986 is an even number, so because there's only one even prime, the total sum will gibe you an odd result

obviously... sum is odd so not divisible by any of the given number. But out of curiosity to know the exact sum and divisibility, i wrote a computer program in Python to perfectly calculate it. You know ... the sum is 1006828939 which is divisible by 1109

all the prime numbers are odd except 2 thus we need odd times prime number to sum up odd. as 2 +odd+odd equals an even.

The last digit of all prime numbers except two are odd. There are 13986 prime numbers including 2. If I exclude 2, there will be only 13985 prime number. Odd number + Odd number = even + odd = odd again, and so on. So, the sum of the first 13985 prime numbers after number 2 is an odd number. Then we add 2, and it will remain odd.

[IMPORTANT, I'M STUCK] Please someone visit my note: https://brilliant.org/discussions/thread/how-to-solve-this-3/

Any Prime number other than 2 and 3 are in the form of 6n+1 or 6n-1.

Case 1: n is even : Sum of first n primes is 2+3+12N = 12N+5

Case 2 : n is odd : Sum is 5+(6n+1 or 6n-1)= 6N+4 or 6N+6

So its obvious that there isn't a common divisor here.