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If N is the sum of the first 13,986 prime numbers, then N is always divisible by:
(a) 6
(b) 4
(c) 8
(d) None of these

seo-qna
Last updated date: 29th Mar 2024
Total views: 394.8k
Views today: 7.94k
MVSAT 2024
Answer
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Hint: to solve this question, we will first find out what even numbers are and what odd numbers are. Then we will take up the cases according to the first n prime numbers where we will vary the value of n. We will see what is the sum obtained when the value of n is even and when it is off. With the help of this, we will determine whether N will be divisible by any of the numbers given in the options.

Complete step by step answer:
Before we solve the question, we will find out what an even number is and what an odd number is. Because we are going to use their concepts further in the solution. Even numbers are those numbers which when divided by 2 leaves the remainder zero. Even numbers end with 0, 2, 4, 6, 8. Odd numbers are those numbers which when divided by 2 leaves the remainder one. Odd numbers end with 1, 3, 5, 7, 9. Now, we will consider first n prime numbers where we will vary the value of n in each case we take up.
Case 1: n = 1.
The number of prime numbers we take is 1. The first prime number is 2. The sum of first n = 1 prime numbers is 2.
Case 2: n = 2.
The number of prime numbers we take is 2. First two prime numbers are 2 and 3. The sum of the first n = 2 prime numbers is 2 + 3 = 5.
Case 3: n = 3.
The number of prime numbers we take is 3. First, three prime numbers are 2, 3, and 5. The sum of the first n = 3 prime numbers is 2 + 3 + 5= 10.
Case 4: n = 4.
The number of prime numbers we take is 4. First, four prime numbers are 2, 3, 5, and 7. The sum of the first n = 4 prime numbers is 2 + 3 + 5 + 7 = 17.
Here, we can see that when we take an odd number of the prime numbers, their sum is even and when we take an even number of prime numbers, their sum is odd. So, we can say that when we take the first 13986 prime numbers, where 13986 is even, then the sum will be odd. Now, we know that if a number is odd, it is not divisible by 2. Thus, the sum will not be divisible by 2.
Now, we can see that in options (a), (b), and (c), we have even numbers and for divisibility by an even number. The even number should be divisible by 2 at least.
In our case, the sum of the first 13986 prime numbers is not divisible by 2, so it will not be divisible by any of the numbers in the options (a), (b) or (c).
Hence, option (d) is the right answer.

Note: The alternate method of solving the question is shown below. In the first 13986 prime numbers, one prime number is 2 and 13985 prime numbers are odd. And we know that when we take the odd number of odd numbers, their sum will be an odd number. Thus the sum of 13985 odd prime numbers will be odd and when we will add 2 in it, we will get an odd number. Thus, it will be not divisible by 4, 6, or 8.