Find the greatest positive integer that will divide 434 and 539 leaving remainders 9 and 12 respectively .

Answer

To find the greatest positive integer that will divide 434 and 539 leaving remainders 9 and 12 respectively, we can follow these steps:

### Step 1: Adjust the numbers by subtracting the remainders
We need to subtract the remainders from the numbers to find the numbers that should be completely divisible by our required integer.

- For 434, subtract the remainder 9:
\[ 434 - 9 = 425 \]
- For 539, subtract the remainder 12:
\[ 539 - 12 = 527 \]

### Step 2: Find the factors of the adjusted numbers
Next, we need to find the factors of both 425 and 527 to identify any common factors.

- **Factors of 425**:
\[ 425 = 5 \times 5 \times 17 = 5^2 \times 17 \]
- **Factors of 527**:
\[ 527 = 17 \times 31 \]

### Step 3: Identify the common factors
Now we look for the common factors between 425 and 527. From our factorization:

- The factors of 425 are \(1, 5, 17, 25, 85, 425\).
- The factors of 527 are \(1, 17, 31, 527\).

The common factor is \(17\).

### Step 4: Conclusion
The greatest positive integer that divides both 434 and 539 leaving remainders 9 and 12 respectively is:
\[ \boxed{17} \]

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