To solve the problem, we need to analyze the number formed by writing one digit six times. Let's denote the digit as \( d \). The number formed by repeating the digit \( d \) six times can be expressed mathematically as:
\[
N = d \times 111111
\]
Now, we need to find out the divisibility of \( N \) by the given options: 7, 11, 13, or all of these.
### Step 1: Understanding the number 111111
The number \( 111111 \) can be expressed as:
\[
111111 = 100000 + 10000 + 1000 + 100 + 10 + 1
\]
### Step 2: Factorizing 111111
To check the divisibility of \( 111111 \), we can factor it. We can also express \( 111111 \) in terms of powers of 10:
\[
111111 = \frac{10^6 - 1}{9}
\]
This is because \( 10^6 - 1 \) gives us \( 999999 \), which is \( 9 \times 111111 \).
### Step 3: Finding the prime factorization of \( 111111 \)
Now, we need to find the prime factors of \( 111111 \):
1. First, we can check if \( 111111 \) is divisible by \( 3 \):
- The sum of the digits of \( 111111 \) is \( 1 + 1 + 1 + 1 + 1 + 1 = 6 \), which is divisible by \( 3 \).
- Thus, \( 111111 \) is divisible by \( 3 \).
2. Dividing \( 111111 \) by \( 3 \):
\[
111111 \div 3 = 37037
\]
3. Now we check \( 37037 \):
- The sum of the digits of \( 37037 \) is \( 3 + 7 + 0 + 3 + 7 = 20 \), which is not divisible by \( 3 \).
- Check for \( 7 \):
\[
37037 \div 7 = 5289
\]
4. Next, we check \( 5289 \):
- The sum of the digits is \( 5 + 2 + 8 + 9 = 24 \), which is divisible by \( 3 \).
- Dividing \( 5289 \) by \( 3 \):
\[
5289 \div 3 = 1763
\]
5. Now we check \( 1763 \):
- Check for \( 13 \):
\[
1763 \div 13 = 135
\]
6. Finally, we check \( 135 \):
- \( 135 = 3 \times 45 \) and \( 45 = 3 \times 15 \) and \( 15 = 3 \times 5 \).
### Final Factorization
Thus, the complete factorization of \( 111111 \) is:
\[
111111 = 3^3 \times 7 \times 13
\]
### Conclusion
Since \( N = d \times 111111 \), and \( 111111 \) is divisible by \( 7 \), \( 11 \), and \( 13 \), we can conclude that the number formed by writing one digit six times is always divisible by \( 7 \), \( 11 \), and \( 13 \).
Thus, the answer is:
**All of these (7, 11, 13)**.
