Explanation:

P is obtained by writing the first 1000 natural numbers from left to right.

We need last four digits of the number formed by the first thousand digits to check the divisibility of 16.

The first 99 natural numbers wll give us 9+(90×2) = 189 digits.

We need another 1000−189 = 811

digits from the remaining numbers numbers.
⇒ required additional natural numbers = 8113 = 270 (approx)

Hence, option 4 is the correct answer.
i.e. 99+270+ one digit
i.e. first  369 natural numbers and one additional digit will form a number containing the first 1000 digits of the number P
i.e. end of the required number will look like .........................3683693

⇒ remainder will be same as the remainder of 3693/16=13

Hence, option 4 is the correct answer.