111,111,111,111 is divisible by

Question

Accepted Answer

To determine the divisibility of the number $$111,111,111,111$$, we analyze its properties based on divisibility rules.

### Steps:

**1. Divisibility by 3:**  
   The sum of all digits in $$111,111,111,111$$ is:  
   $$
   1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 12
   $$  
   Since $$12$$ is divisible by $$3$$, the number $$111,111,111,111$$ is divisible by $$3$$.

**2. Divisibility by 11:**  
   The difference between the sum of digits at odd places and even places is:  
   $$
   (Sum \ of \ odd \ places) - (Sum \ of \ even \ places) = 6 - 6 = 0
   $$  
   Since the result is $$0$$ (divisible by $$11$$), the number is divisible by $$11$$.

**3. Divisibility by 37:**  
   When divided by $$37$$, the number gives:  
   $$
   111,111,111,111 \div 37 = 3,003,003,003
   $$  
   Hence, it is divisible by $$37$$.

**4. Divisibility by 111:**  
   When divided by $$111$$:  
   $$
   111,111,111,111 \div 111 = 1,001,001,001
   $$  
   Thus, it is divisible by $$111$$.

**5. Divisibility by 1001:**  
   Similarly:  
   $$
   111,111,111,111 \div 1001 = 111,000
   $$  
   Hence it is divisible by $$1001$$.

### Final Answer:  
The number $$111,111,111,111$$ is divisible by **3**, **11**, **37**, **111**, and **1001**.

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