The greatest integer that divides 358, 376, 232 leaving the same remiander 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: Understand the Problem
We need to find a number \( d \) such that when we divide 358, 376, and 232 by \( d \), we get the same remainder \( r \). This means that the differences between the numbers must be divisible by \( d \).

### Step 2: Calculate the Differences
We will calculate the differences between the pairs of numbers:

1. Difference between 376 and 358:
\[ 376 - 358 = 18 \]

2. Difference between 376 and 232:
\[ 376 - 232 = 144 \]

3. Difference between 358 and 232:
\[ 358 - 232 = 126 \]

### Step 3: Find the GCD of the Differences
Now, we need to find the greatest common divisor (GCD) of the differences we calculated: 18, 144, and 126.

1. **Finding GCD of 18 and 144:**
- 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 GCD is 18.

2. **Finding GCD of 18 and 126:**
- The factors of 126 are: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
- The common factors with 18 are: 1, 2, 3, 6, 9, 18
- The GCD remains 18.

Thus, the GCD of 18, 144, and 126 is 18.

### Step 4: Verify the Result
To ensure that 18 is the correct answer, we can check the remainders when we divide each number by 18:

1. \( 358 \div 18 = 19 \) remainder \( 16 \)
2. \( 376 \div 18 = 20 \) remainder \( 16 \)
3. \( 232 \div 18 = 12 \) remainder \( 16 \)

Since all three divisions give the same remainder of 16, we confirm that our answer is correct.

### Final Answer
The greatest integer that divides 358, 376, and 232 leaving the same remainder in each case is **18**.

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