The number 311 311 311 311 311 311 311 is

Options

A. Divisible by both 3 and 11
B. Divisible by 3 but not by 11 Divisibel by 11 but not by11
C. Divisible by 11 but not by 3
D. Neither divisible by 3 nor by 11.

Answer

To determine whether the number 311311311311311311 is divisible by 3 and 11, we will follow these steps:

### Step 1: Check divisibility by 3
To check if a number is divisible by 3, we need to sum all its digits and see if that sum is divisible by 3.

**Digits of the number:** 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1

**Count of digits:**
- The digit '3' appears 7 times.
- The digit '1' appears 11 times.

**Sum of digits:**
- Sum = (3 * 7) + (1 * 11)
- Sum = 21 + 11 = 32

**Check divisibility:**
- Now, we check if 32 is divisible by 3.
- 32 ÷ 3 = 10.67 (not an integer)

Since 32 is not divisible by 3, the number is **not divisible by 3**.

### Step 2: Check divisibility by 11
To check if a number is divisible by 11, we find the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions.

**Odd position digits:**
- 3, 1, 1, 3, 1, 1, 3, 1, 1 (1st, 3rd, 5th, 7th, 9th, 11th, 13th, 15th, 17th)
- Sum of odd position digits = 3 + 1 + 1 + 3 + 1 + 1 + 3 + 1 + 1 = 16

**Even position digits:**
- 1, 3, 1, 3, 1, 3, 1, 3 (2nd, 4th, 6th, 8th, 10th, 12th, 14th, 16th)
- Sum of even position digits = 1 + 3 + 1 + 3 + 1 + 3 + 1 + 3 = 16

**Difference:**
- Difference = |Sum of odd position - Sum of even position|
- Difference = |16 - 16| = 0

**Check divisibility:**
- 0 is divisible by 11.

Since the difference is divisible by 11, the number is **divisible by 11**.

### Conclusion
- The number 311311311311311311 is **not divisible by 3** but **is divisible by 11**.

### Final Answer
The correct option is **C: divisible by 11 but not by 3**.

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