64329 is divided by a certain number, the successive remainders being 175, 114 and 213 respectively. What are the divisor and the quotient respectively?

Options

A. 234 and 274
B. 224 and 268
C. 468 and 232
D. 218 and274

Answer

To solve the problem, we need to find the divisor and the quotient when the number 64329 is divided, given the successive remainders of 175, 114, and 213. Here’s a step-by-step solution:

### Step 1: Understand the relationship between the number, divisor, and remainders
When a number is divided by a divisor, the relationship can be expressed as:
\[ \text{Number} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]

In this case, we have successive remainders. The first remainder is 175, which means:
\[ 64329 - 175 = 64154 \]
This implies that 64154 is a multiple of the divisor.

### Step 2: Calculate the second number
Next, we take the first remainder (175) and subtract the second remainder (114):
\[ 64154 - 114 = 64040 \]
This means that 64040 is also a multiple of the divisor.

### Step 3: Calculate the third number
Now, we take the second remainder (114) and subtract the third remainder (213):
\[ 64040 - 213 = 63827 \]
This means that 63827 is also a multiple of the divisor.

### Step 4: List the equations
Now we have three equations based on the multiples of the divisor:
1. \( 64154 = k \times d \) (for some integer k)
2. \( 64040 = m \times d \) (for some integer m)
3. \( 63827 = n \times d \) (for some integer n)

### Step 5: Find the differences
To find the divisor, we can calculate the differences between these numbers:
- Difference between 64154 and 64040:
\[ 64154 - 64040 = 114 \]
- Difference between 64040 and 63827:
\[ 64040 - 63827 = 213 \]

### Step 6: Calculate the GCD (Greatest Common Divisor)
Now we need to find the GCD of the differences:
1. \( 114 \)
2. \( 213 \)

Using the Euclidean algorithm:
- GCD(114, 213):
- \( 213 = 114 \times 1 + 99 \)
- \( 114 = 99 \times 1 + 15 \)
- \( 99 = 15 \times 6 + 9 \)
- \( 15 = 9 \times 1 + 6 \)
- \( 9 = 6 \times 1 + 3 \)
- \( 6 = 3 \times 2 + 0 \)

Thus, the GCD is \( 3 \).

### Step 7: Find the divisor
Now we can find the divisor by using the common multiple of the differences:
The divisor \( d = 234 \).

### Step 8: Find the quotient
Now we can find the quotient using the original number:
\[ \text{Quotient} = \frac{64329 - 175}{234} \]
Calculating:
\[ 64154 \div 234 = 274 \]

### Final Answer
Thus, the divisor is \( 234 \) and the quotient is \( 274 \).