Question

If N is the sum of first 13986 prime numbers, then N is always divisible by:
(a) 6
(b) 4
(c) 8
(d) none of these

Answer 1

Hint: First, before proceeding for this, we must know the following conditions that if we add an odd number with the odd number three times, we always get an odd number as a result. Then, the sum of the 13986 terms will give the result as 2+ sum of 13985 other prime numbers which are odd in nature. Then, the sum will always be an odd number, which proves that it can’t be divisible by any even number which gives our final answer.

Complete step-by-step answer:
In this question, we are supposed to find if N is the sum of first 13986 prime numbers, then N is always divisible by which number.
So, before proceeding for this, we must know the following conditions: if we add an odd number with the odd number three times, we always get an odd number as a result.
To understand it take an example of 3, 5 and 7 as 3+5+7=15 which is also an odd number.
So, in this question the first prime number is 2, which is even in nature but still a prime number as 2 is the only even prime number we have and the other rest all prime numbers are odd.
So, the sum of the 13986 terms will give the result as 2+ sum of 13985 other prime numbers which are odd in nature.
Then, we also know the fact that the sum of even and odd numbers always gives an odd number.
So, the sum will always be an odd number, which proves that it can’t be divisible by any even number.
Then, in the question we can see that all the options given are even and the sum of prime numbers can’t be divisible by even numbers.
So, none of the given choices is correct which proves the answer is none of these.
So, the correct answer is “Option D”.

Note: Now, to solve these types of questions we need to know some of the basic rules and definitions so to get the answer easily. Here, definition of prime number is important which states the number which is divisible by itself or 1 is called prime number. Moreover, 2 is the only even prime number.