N is a five digit number. 1 is written after the 5 digit of N to make it a six digit number, which is three times the same number with 1 written before N. (If N = 23456 it means 234561 and 123456). Then the middle digit of the number N is

Answer

To solve the problem step by step, we need to find a five-digit number \( N \) such that when we write 1 after \( N \), it becomes a six-digit number equal to three times the number formed by writing 1 before \( N \).

### Step 1: Define the numbers
Let \( N \) be the five-digit number. When we write 1 after \( N \), we can express this as:
\[ 10N + 1 \]
When we write 1 before \( N \), we can express this as:
\[ 100000 + N \]

### Step 2: Set up the equation
According to the problem, the six-digit number \( 10N + 1 \) is equal to three times the number \( 100000 + N \). Therefore, we can set up the equation:
\[ 10N + 1 = 3(100000 + N) \]

### Step 3: Expand and simplify the equation
Expanding the right side gives us:
\[ 10N + 1 = 300000 + 3N \]
Now, we can rearrange the equation to isolate \( N \):
\[ 10N - 3N = 300000 - 1 \]
This simplifies to:
\[ 7N = 299999 \]

### Step 4: Solve for \( N \)
Now we can solve for \( N \):
\[ N = \frac{299999}{7} \]
Calculating this gives:
\[ N = 42857 \]

### Step 5: Find the middle digit
The five-digit number \( N \) is \( 42857 \). The middle digit of this number is the third digit, which is:
\[ 8 \]

Thus, the middle digit of the number \( N \) is \( 8 \).

### Final Answer
The middle digit of the number \( N \) is \( 8 \).

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