Find the greatest number which divides 99, 123 and 183 leaving the same remainder in each case

Options

A. 11
B. 12
C. 13
D. 14

Answer

To find the greatest number that divides 99, 123, and 183 while leaving the same remainder in each case, we can follow these steps:

### Step 1: Identify the numbers
We have three numbers:
- a = 99
- b = 123
- c = 183

### Step 2: Calculate the differences between the numbers
We need to find the differences between the pairs of numbers:
1. \( b - a = 123 - 99 = 24 \)
2. \( c - b = 183 - 123 = 60 \)
3. \( c - a = 183 - 99 = 84 \)

### Step 3: List the differences
Now we have the three differences:
- 24
- 60
- 84

### Step 4: Find the HCF of the differences
Next, we need to find the highest common factor (HCF) of the differences (24, 60, and 84).

1. **Finding HCF of 24 and 60:**
- The factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- The factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- Common factors: 1, 2, 3, 4, 6, 12
- HCF(24, 60) = 12

2. **Finding HCF of 12 and 84:**
- The factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
- Common factors with 12: 1, 2, 3, 4, 6, 12
- HCF(12, 84) = 12

### Step 5: Conclusion
The greatest number that divides 99, 123, and 183 leaving the same remainder is **12**.

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