The numbers 2270 , 3739 and 6677 on being divided by a certain number of three digits, leave the same remainder. Find the number and the remainder respectively

Options

A. 123, 20
B. 113, 10
C. 116, 20
D. 118, 15

Answer

To solve the problem, we need to find a three-digit number that leaves the same remainder when dividing the numbers 2270, 3739, and 6677. Here’s a step-by-step solution:

### Step 1: Find the differences between the numbers
We start by calculating the differences between the given numbers. This is important because if all three numbers leave the same remainder when divided by a certain number, then the differences between the numbers must be divisible by that number.

1. **Difference between 3739 and 2270:**
\[ 3739 - 2270 = 1469 \]

2. **Difference between 6677 and 3739:**
\[ 6677 - 3739 = 2938 \]

3. **Difference between 6677 and 2270:**
\[ 6677 - 2270 = 4407 \]

### Step 2: Find the HCF of the differences
Next, we need to find the highest common factor (HCF) of the differences we calculated: 1469, 2938, and 4407.

1. **Finding HCF of 1469 and 2938:**
- Factor 1469: It is a prime number.
- Factor 2938: \(2938 = 1469 \times 2\)
- HCF(1469, 2938) = 1469

2. **Finding HCF of 1469 and 4407:**
- Factor 4407: \(4407 = 1469 \times 3\)
- HCF(1469, 4407) = 1469

Thus, the HCF of 1469, 2938, and 4407 is **1469**.

### Step 3: Find a three-digit factor of the HCF
Now, we need to find a three-digit number that is a factor of 1469. The prime factorization of 1469 is:
\[ 1469 = 113 \times 13 \]
The three-digit number we can take from this factorization is **113**.

### Step 4: Calculate the remainder
Now we will calculate the remainder when we divide each of the original numbers by 113 to confirm they all give the same remainder.

1. **For 2270:**
\[ 2270 \div 113 = 20 \quad \text{(quotient)}, \quad 2270 - (113 \times 20) = 10 \quad \text{(remainder)} \]

2. **For 3739:**
\[ 3739 \div 113 = 33 \quad \text{(quotient)}, \quad 3739 - (113 \times 33) = 10 \quad \text{(remainder)} \]

3. **For 6677:**
\[ 6677 \div 113 = 59 \quad \text{(quotient)}, \quad 6677 - (113 \times 59) = 10 \quad \text{(remainder)} \]

In all cases, the remainder is **10**.

### Final Answer
The three-digit number is **113** and the remainder is **10**.

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