The number 311311311311311311311 is

Options

A. divisible by 3 but not by 11
B. divisible by 11 but not by 3
C. divisible by both 3 and 11
D. neither divisble by 3 nor by 11

Answer

To determine whether the number 311311311311311311311 is divisible by 3 or 11, we will follow these steps:

### Step 1: Identify the number
The given number is 311311311311311311311.

### Step 2: Check divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3.

- **Calculate the sum of the digits:**
- The digits are: 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1
- Grouping the digits: (3 + 1 + 1) = 5
- The sequence "311" appears 7 times in the number.

- **Total sum of the digits:**
- Sum = 5 × 7 = 35

- **Check if 35 is divisible by 3:**
- 35 ÷ 3 = 11 remainder 2
- Since there is a remainder, 35 is not divisible by 3.

### Step 3: Check divisibility by 11
A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is divisible by 11.

- **Identify the positions of the digits:**
- Odd positions: 1st, 3rd, 5th, 7th, 9th, 11th, 13th, 15th, 17th, 19th
- Even positions: 2nd, 4th, 6th, 8th, 10th, 12th, 14th, 16th, 18th, 20th

- **Calculate the sum of the digits in odd positions:**
- Odd positions: 3, 1, 1, 3, 1, 1, 3, 1, 1
- Sum of odd positions = 3 + 1 + 1 + 3 + 1 + 1 + 3 + 1 + 1 = 18

- **Calculate the sum of the digits in even positions:**
- Even positions: 1, 3, 1, 3, 1, 3, 1, 3, 1
- Sum of even positions = 1 + 3 + 1 + 3 + 1 + 3 + 1 + 3 + 1 = 16

- **Calculate the difference:**
- Difference = Sum of odd positions - Sum of even positions
- Difference = 18 - 16 = 2

- **Check if 2 is divisible by 11:**
- 2 ÷ 11 = 0 remainder 2
- Since there is a remainder, 2 is not divisible by 11.

### Conclusion
Since the number 311311311311311311311 is neither divisible by 3 nor by 11, the correct option is:
- Neither divisible by 3 nor by 11.

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