The first twenty natural numbers from 1 to 20 are written next to each other to form a 31 digit number `{:(N,=,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20):}`. What is the remainder when this number is divided by 16?

Options

A. 0
B. 4
C. 7
D. 9

Answer

To solve the problem of finding the remainder when the 31-digit number formed by writing the first twenty natural numbers (1 to 20) next to each other is divided by 16, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Number**:
The first twenty natural numbers written next to each other form the number:
\( 1234567891011121314151617181920 \).

2. **Focus on the Last Four Digits**:
To find the remainder when this number is divided by 16, we only need to consider the last four digits of the number. The last four digits of our number are:
\( 1920 \).

3. **Divide the Last Four Digits by 16**:
Now, we will divide \( 1920 \) by \( 16 \) to find the remainder:
\[ 1920 \div 16 \]

4. **Perform the Division**:
- First, calculate how many times \( 16 \) fits into \( 1920 \):
\[ 16 \times 120 = 1920 \]
- Since \( 1920 \) is exactly divisible by \( 16 \), there is no remainder.

5. **Conclusion**:
Since \( 1920 \) is divisible by \( 16 \), the remainder when the entire number is divided by \( 16 \) is:
\[ \text{Remainder} = 0 \]

### Final Answer:
The remainder when the number \( 1234567891011121314151617181920 \) is divided by \( 16 \) is \( 0 \).

---