The largest number, which divides 25, 73 and 97 to leave the same remainder in each case,is

Options

A. 24
B. 23
C. 21
D. 6

Answer

To find the largest number that divides 25, 73, and 97 leaving the same remainder, we can follow these steps:

### Step 1: Understand the Concept
We need to find a number \( x \) such that when we subtract \( x \) from each of the numbers (25, 73, and 97), the results are all divisible by \( x \). This means we need to find the highest common factor (HCF) of the differences between the numbers.

### Step 2: Calculate the Differences
First, we calculate the differences between the numbers:
1. Difference between 73 and 25:
\[ 73 - 25 = 48 \]
2. Difference between 97 and 73:
\[ 97 - 73 = 24 \]
3. Difference between 97 and 25:
\[ 97 - 25 = 72 \]

### Step 3: List the Differences
We have the following differences:
- 48
- 24
- 72

### Step 4: Find the HCF of the Differences
Now, we need to find the HCF of 48, 24, and 72.

1. **Finding the factors:**
- Factors of 48: \( 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 \)
- Factors of 24: \( 1, 2, 3, 4, 6, 8, 12, 24 \)
- Factors of 72: \( 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 \)

2. **Common factors:**
The common factors of 48, 24, and 72 are \( 1, 2, 3, 4, 6, 8, 12, 24 \).

3. **Finding the highest common factor:**
The highest common factor is \( 24 \).

### Step 5: Conclusion
Thus, the largest number that divides 25, 73, and 97 leaving the same remainder is **24**.

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