The number 1563241234351 is:

Options

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

Answer

To determine whether the number 1563241234351 is divisible by 11 or 3, we will follow the divisibility rules for both numbers step by step.

### Step 1: Check Divisibility by 3
The rule for divisibility by 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3.

**Calculation of the sum of the digits:**
- Break down the number into its individual digits: 1, 5, 6, 3, 2, 4, 1, 2, 3, 4, 3, 5, 1.
- Now, sum these digits:
- \(1 + 5 = 6\)
- \(6 + 6 = 12\)
- \(12 + 3 = 15\)
- \(15 + 2 = 17\)
- \(17 + 4 = 21\)
- \(21 + 1 = 22\)
- \(22 + 2 = 24\)
- \(24 + 3 = 27\)
- \(27 + 4 = 31\)
- \(31 + 3 = 34\)
- \(34 + 5 = 39\)
- \(39 + 1 = 40\)

**Final sum of digits:** 40

**Check divisibility by 3:**
- Since 40 is not divisible by 3, the number 1563241234351 is **not divisible by 3**.

### Step 2: Check Divisibility by 11
The rule for divisibility by 11 states that 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 either 0 or divisible by 11.

**Identify the positions of the digits:**
- Odd positions: 1st, 3rd, 5th, 7th, 9th, 11th
- Even positions: 2nd, 4th, 6th, 8th, 10th, 12th

**Sum of digits in odd positions:**
- Odd positions: 1 (1st), 6 (3rd), 2 (5th), 2 (7th), 3 (9th), 1 (11th)
- Sum = \(1 + 6 + 2 + 2 + 3 + 1 = 15\)

**Sum of digits in even positions:**
- Even positions: 5 (2nd), 3 (4th), 4 (6th), 1 (8th), 4 (10th), 5 (12th)
- Sum = \(5 + 3 + 4 + 1 + 4 + 5 = 22\)

**Calculate the difference:**
- Difference = \( |15 - 22| = 7 \)

**Check divisibility by 11:**
- Since 7 is not equal to 0 and not divisible by 11, the number 1563241234351 is **not divisible by 11**.

### Conclusion
The number 1563241234351 is neither divisible by 3 nor by 11.

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