111111111111 is divisible by :

Options

A. a)3 and 37 only
B. b)3,37 and 11 only
C. c)3,11, 37 and 111 only
D. d)3,11,37,111 and 1001

Answer

To determine the divisibility of the number 111111111111, we can follow these steps:

### Step 1: Understand the structure of the number
The number 111111111111 consists of the digit '1' repeated 12 times.

### Step 2: Recognize the divisibility rule for repeated digits
A number that consists of a single digit repeated \( n \) times can be expressed as:
\[ \text{Number} = d \times \frac{10^n - 1}{9} \]
where \( d \) is the repeated digit and \( n \) is the number of times it is repeated. In our case, \( d = 1 \) and \( n = 12 \).

### Step 3: Calculate the expression
For our number:
\[ 111111111111 = 1 \times \frac{10^{12} - 1}{9} \]

### Step 4: Identify the factors of \( \frac{10^{12} - 1}{9} \)
The number \( 10^{12} - 1 \) can be factored using the difference of squares and other algebraic identities:
\[ 10^{12} - 1 = (10^6 - 1)(10^6 + 1) \]
Continuing to factor \( 10^6 - 1 \):
\[ 10^6 - 1 = (10^3 - 1)(10^3 + 1) \]
And further factoring \( 10^3 - 1 \):
\[ 10^3 - 1 = (10 - 1)(10^2 + 10 + 1) = 9 \times 111 \]
Thus, we have:
\[ 10^{12} - 1 = (9 \times 111)(10^6 + 1) \]

### Step 5: Determine the divisibility
From the factors, we can see that:
- \( 9 \) is a factor (which means it is divisible by \( 3 \)).
- \( 111 \) can be factored into \( 3 \times 37 \), meaning it is also divisible by \( 3 \) and \( 37 \).
- The number \( 10^6 + 1 \) is not divisible by \( 3 \), \( 7 \), \( 11 \), or \( 13 \).

### Conclusion
Thus, the number 111111111111 is divisible by:
- \( 3 \)
- \( 37 \)
- \( 111 \)

### Final Answer
The number 111111111111 is divisible by \( 3, 37, \) and \( 111 \).

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