An eight digit number is a multiple of 73 and 137. If the second digit from left is 7, what is the 6th digit from the left of the number ?

Explanation

To solve the problem step by step, we need to find an eight-digit number that is a multiple of both 73 and 137, with the condition that the second digit from the left is 7. We will also determine the sixth digit from the left of this number.

### Step 1: Find the product of 73 and 137
First, we need to calculate the product of 73 and 137 to find the least common multiple (LCM) of these two numbers.

\[ 73 \times 137 = 10001 \]

### Step 2: Identify the eight-digit multiples of 10001
Since we are looking for an eight-digit number, we need to find multiples of 10001 that result in an eight-digit number. The smallest eight-digit number is 10,000,000.

To find the smallest integer \( n \) such that \( 10001n \) is at least 10,000,000, we can set up the inequality:

\[ 10001n \geq 10,000,000 \]

Dividing both sides by 10001 gives:

\[ n \geq \frac{10,000,000}{10001} \approx 999.9 \]

Thus, the smallest integer \( n \) is 1000.

### Step 3: Generate eight-digit multiples of 10001
Now we will generate the multiples of 10001 starting from \( n = 1000 \):

- For \( n = 1000 \):
\[ 10001 \times 1000 = 10,001,000 \]
- For \( n = 1001 \):
\[ 10001 \times 1001 = 10,011,001 \]
- For \( n = 1002 \):
\[ 10001 \times 1002 = 10,021,002 \]
- For \( n = 1003 \):
\[ 10001 \times 1003 = 10,031,003 \]
- For \( n = 1004 \):
\[ 10001 \times 1004 = 10,041,004 \]

Continuing this process, we can find several eight-digit numbers.

### Step 4: Check the second digit
We need to check which of these numbers has the second digit as 7:

- \( 10,001,000 \) - second digit is 0
- \( 10,011,001 \) - second digit is 0
- \( 10,021,002 \) - second digit is 0
- \( 10,031,003 \) - second digit is 0
- \( 10,041,004 \) - second digit is 0
- \( 10,051,005 \) - second digit is 0
- \( 10,061,006 \) - second digit is 0
- \( 10,071,007 \) - second digit is 0
- \( 10,081,008 \) - second digit is 0
- \( 10,091,009 \) - second digit is 0
- \( 10,101,010 \) - second digit is 0
- \( 10,111,011 \) - second digit is 0
- \( 10,121,012 \) - second digit is 0
- \( 10,131,013 \) - second digit is 0
- \( 10,141,014 \) - second digit is 0
- \( 10,151,015 \) - second digit is 0
- \( 10,161,016 \) - second digit is 0
- \( 10,171,017 \) - second digit is 7

### Step 5: Find the sixth digit
Now we have found that \( 10,171,017 \) has the second digit as 7. The sixth digit from the left is:

\[ 10,171,017 \Rightarrow \text{6th digit is } 0 \]

### Conclusion
Thus, the sixth digit from the left of the eight-digit number is **0**.