Find the greatest number that will divide 47, 95 and 187 so as to leave the same remainder in each case.

Options

A. 7
B. 13
C. 9
D. 4

Answer

To find the greatest number that will divide 47, 95, and 187 leaving the same remainder, we can follow these steps:

### Step 1: Identify the numbers
We have three numbers: 47, 95, and 187.

### Step 2: Calculate the differences between the numbers
To find a number that leaves the same remainder when dividing these three numbers, we first calculate the differences between the pairs of numbers:
- Difference between 187 and 95:
\[ 187 - 95 = 92 \]
- Difference between 95 and 47:
\[ 95 - 47 = 48 \]
- Difference between 187 and 47:
\[ 187 - 47 = 140 \]

### Step 3: List the differences
Now we have the differences:
- \( 92 \)
- \( 48 \)
- \( 140 \)

### Step 4: Find the HCF of the differences
Next, we need to find the highest common factor (HCF) of these differences: 92, 48, and 140.

#### Finding HCF of 92 and 48:
1. Prime factorization of 92:
\[ 92 = 2^2 \times 23 \]
2. Prime factorization of 48:
\[ 48 = 2^4 \times 3 \]
3. Common factors: The only common prime factor is \( 2 \).
4. The lowest power of \( 2 \) is \( 2^2 \).
5. Therefore, HCF of 92 and 48 is \( 4 \).

#### Finding HCF of 4 and 140:
1. Prime factorization of 140:
\[ 140 = 2^2 \times 5 \times 7 \]
2. Common factors with 4: The common prime factor is \( 2 \).
3. The lowest power of \( 2 \) is \( 2^2 \).
4. Therefore, HCF of 4 and 140 is \( 4 \).

### Step 5: Conclusion
The greatest number that will divide 47, 95, and 187 leaving the same remainder is:
\[ \text{HCF} = 4 \]

### Final Answer
The answer is \( 4 \).

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