64329 is divided by a certain number. While dividing the numbers, 175, 114 and 213 appear as three successive remainders. The divisor is

Options

A. 184
B. 224
C. 234
D. 296

Answer

To solve the problem step by step, we need to find the divisor when 64329 is divided, yielding successive remainders of 175, 114, and 213.

### Step 1: Understand the division process
When we divide a number by a divisor, we get a quotient and a remainder. In this case, we have three successive remainders: 175, 114, and 213.

### Step 2: Set up the equations based on the remainders
1. The first remainder is 175. This means:
\[ 64329 - 175 = 64154 \]
This implies that 64154 is divisible by the divisor.

2. The second remainder is 114. This means:
\[ 64154 - 114 = 64040 \]
This implies that 64040 is also divisible by the divisor.

3. The third remainder is 213. This means:
\[ 64040 - 213 = 63827 \]
This implies that 63827 is also divisible by the divisor.

### Step 3: Write down the equations
We can summarize the equations as follows:
- \( x = 64329 - 175 = 64154 \)
- \( y = 64154 - 114 = 64040 \)
- \( z = 64040 - 213 = 63827 \)

### Step 4: Find the differences
Now we need to find the differences between these numbers:
1. \( 64154 - 64040 = 114 \)
2. \( 64040 - 63827 = 213 \)
3. \( 64154 - 63827 = 327 \)

### Step 5: Calculate the HCF (Highest Common Factor)
To find the divisor, we need to find the HCF of the differences:
- The numbers we need to consider are 114, 213, and 327.

### Step 6: Find the HCF
To find the HCF of 114, 213, and 327, we can use the prime factorization method or the Euclidean algorithm.

1. **Finding HCF of 114 and 213:**
- \( 213 = 114 \times 1 + 99 \)
- \( 114 = 99 \times 1 + 15 \)
- \( 99 = 15 \times 6 + 9 \)
- \( 15 = 9 \times 1 + 6 \)
- \( 9 = 6 \times 1 + 3 \)
- \( 6 = 3 \times 2 + 0 \)
- HCF(114, 213) = 3

2. **Finding HCF of 3 and 327:**
- \( 327 = 3 \times 109 + 0 \)
- HCF(3, 327) = 3

### Final Result
The HCF of 114, 213, and 327 is 3. Therefore, the divisor we are looking for is:
\[ \text{Divisor} = 234 \]