On dividing the numbers 7654, 8506 and 9997 by a certain largest number ,in each case the remainder is the same find the number and the remainder

Options

A. 213 and 199
B. 223 and 189
C. 233 and 179
D. None of these

Answer

To solve the problem of finding the largest number that leaves the same remainder when dividing the numbers 7654, 8506, and 9997, we can follow these steps:

### Step 1: Find the differences between the numbers
First, we will calculate the differences between the pairs of numbers:

1. Difference between 8506 and 7654:
\[ 8506 - 7654 = 852 \]

2. Difference between 9997 and 8506:
\[ 9997 - 8506 = 1491 \]

3. Difference between 9997 and 7654:
\[ 9997 - 7654 = 2343 \]

### Step 2: Calculate the HCF of the differences
Next, we need to find the Highest Common Factor (HCF) of the differences we calculated:

- The differences are 852, 1491, and 2343.

To find the HCF, we can use the prime factorization or the Euclidean algorithm. Here, we will use the Euclidean algorithm:

1. Find HCF of 852 and 1491:
- Divide 1491 by 852:
\[ 1491 \div 852 = 1 \quad \text{(remainder: 639)} \]
- Now, find HCF of 852 and 639:
\[ 852 \div 639 = 1 \quad \text{(remainder: 213)} \]
- Now, find HCF of 639 and 213:
\[ 639 \div 213 = 3 \quad \text{(remainder: 0)} \]
- So, HCF(852, 1491) = 213.

2. Now find HCF of 213 and 2343:
- Divide 2343 by 213:
\[ 2343 \div 213 = 11 \quad \text{(remainder: 0)} \]
- So, HCF(213, 2343) = 213.

Thus, the HCF of 852, 1491, and 2343 is **213**.

### Step 3: Find the remainder
Now that we have the largest number (HCF = 213), we can find the remainder when we divide any of the original numbers by this HCF.

Let's divide 8506 by 213:
\[ 8506 \div 213 = 39 \quad \text{(remainder: 199)} \]

### Conclusion
The largest number that divides 7654, 8506, and 9997 leaving the same remainder is **213**, and the remainder when dividing these numbers by 213 is **199**.