To determine whether the number \( 311311311311311311311 \) is divisible by 3 or 11, we need to apply the divisibility rules for these numbers.

Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

1. Identify the digits of the number: In \( 311311311311311311311 \), the digits are repeated sequences of 3, 1, and 1.
2. Count and sum the digits:
   • 3 is repeated 7 times.
   • 1 is repeated 14 times.
   • Sum of all digits = \( 7 \times 3 + 14 \times 1 = 21 + 14 = 35 \).
3. Check divisibility: 35 is not divisible by 3 (since \( 35 \div 3 \) leaves a remainder).

Thus, the number is not divisible by 3.

Divisibility by 11

A number is divisible by 11 if the difference between the sum of the digits at odd positions and the sum of the digits at even positions is either 0 or divisible by 11.

1. Split the number into odd and even positions:
   • Odd positions: 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3
   • Even positions: 1, 3, 1, 3, 1, 3, 1, 3, 1, 3
2. Calculate the sum of odd and even positions:
   • Sum of odd-positioned digits = \( 22 \) (sum calculated as \( 3+1+3+1+...+3 \)).
   • Sum of even-positioned digits = \( 13 \) (sum calculated as \( 1+3+1+3+...+3 \)).
   • Difference = \( 22 - 13 = 9 \).
3. Check divisibility: 9 is not divisible by 11.

Thus, the number is not divisible by 11.

Conclusion

Since the number \( 311311311311311311311 \) is neither divisible by 3 nor by 11, the correct answer is:

Option: Neither divisible by 3 nor by 11.