The greatest integer that divides 358, 376, 232 leaving the same remainder in each case is:

Options

A. 6
B. 7
C. 8
D. 9

Answer

To find the greatest integer that divides 358, 376, and 232 leaving the same remainder in each case, we can follow these steps:

### Step 1: Set up the equations
Let the greatest integer be \( n \) and the common remainder be \( r \). We can express the three numbers in terms of \( n \) and \( r \):
- \( 358 = n \cdot a + r \)
- \( 376 = n \cdot b + r \)
- \( 232 = n \cdot c + r \)

### Step 2: Subtract the equations
By subtracting the first equation from the second, we eliminate \( r \):
\[ 376 - 358 = n(b - a) \]
This simplifies to:
\[ 18 = n(b - a) \quad \text{(1)} \]

Next, subtract the third equation from the second:
\[ 376 - 232 = n(b - c) \]
This simplifies to:
\[ 144 = n(b - c) \quad \text{(2)} \]

### Step 3: Find the common divisor
From equations (1) and (2), we have:
1. \( n(b - a) = 18 \)
2. \( n(b - c) = 144 \)

To find \( n \), we need to find the greatest common divisor (GCD) of the two results (18 and 144).

### Step 4: Calculate the GCD
The factors of 18 are:
- \( 1, 2, 3, 6, 9, 18 \)

The factors of 144 are:
- \( 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 \)

The common factors are:
- \( 1, 2, 3, 6, 9, 18 \)

The greatest of these common factors is \( 18 \).

### Step 5: Verify the divisibility
Now, we need to check if \( 9 \) divides \( 18 \) and \( 144 \):
- \( 18 \div 9 = 2 \) (which is an integer)
- \( 144 \div 9 = 16 \) (which is also an integer)

Thus, \( n = 9 \) is indeed a divisor.

### Conclusion
The greatest integer that divides 358, 376, and 232 leaving the same remainder in each case is \( \boxed{9} \).

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