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, we need to determine the divisor based on the remainders given when dividing the number 64329. The remainders provided are 175, 114, and 213. Let's break down the solution step-by-step.

### Step 1: Understand the problem
We are given the number 64329 and three successive remainders when it is divided by an unknown divisor. The remainders are 175, 114, and 213.

### Step 2: Set up the equations
When dividing a number, the relationship between the number, the divisor, and the remainder can be expressed as:
\[ \text{Number} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]

From the problem, we can set up the following equations based on the remainders:

1. For the first remainder (175):
\[ 64329 = d \times q_1 + 175 \]
This implies:
\[ 64329 - 175 = d \times q_1 \implies 64154 = d \times q_1 \]

2. For the second remainder (114):
\[ 64154 = d \times q_2 + 114 \]
This implies:
\[ 64154 - 114 = d \times q_2 \implies 64040 = d \times q_2 \]

3. For the third remainder (213):
\[ 64040 = d \times q_3 + 213 \]
This implies:
\[ 64040 - 213 = d \times q_3 \implies 63827 = d \times q_3 \]

### Step 3: Find the differences between the remainders
To find the divisor, we need to look at the differences between the numbers we calculated:

1. The difference between 64154 and 64040:
\[ 64154 - 64040 = 114 \]

2. The difference between 64040 and 63827:
\[ 64040 - 63827 = 213 \]

### Step 4: Calculate the GCD
Now, we need to find the greatest common divisor (GCD) of the differences we calculated:
- The differences are 114 and 213.

To find the GCD of 114 and 213, we can use the Euclidean algorithm:

1. Divide 213 by 114:
\[ 213 = 114 \times 1 + 99 \]
2. Now, divide 114 by 99:
\[ 114 = 99 \times 1 + 15 \]
3. Now, divide 99 by 15:
\[ 99 = 15 \times 6 + 9 \]
4. Now, divide 15 by 9:
\[ 15 = 9 \times 1 + 6 \]
5. Now, divide 9 by 6:
\[ 9 = 6 \times 1 + 3 \]
6. Finally, divide 6 by 3:
\[ 6 = 3 \times 2 + 0 \]

The GCD is the last non-zero remainder, which is 3.

### Conclusion
The divisor is 3.

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