What is the greatest number that will divide 209 and 347 leaving remainder 5 and 7 respectively?

Options

A. 23
B. 16
C. 19
D. 17

Answer

To solve the problem of finding the greatest number that will divide 209 and 347 leaving remainders of 5 and 7 respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find a number \( x \) such that:
- When 209 is divided by \( x \), the remainder is 5.
- When 347 is divided by \( x \), the remainder is 7.

2. **Setting Up the Equations**:
From the conditions given, we can express the two numbers in terms of \( x \):
- \( 209 - 5 = 204 \) must be divisible by \( x \).
- \( 347 - 7 = 340 \) must be divisible by \( x \).

3. **Finding the Numbers to Consider**:
Therefore, we need to find the greatest common divisor (GCD) of 204 and 340.

4. **Using the Euclidean Algorithm**:
We will use the division method (Euclidean algorithm) to find the GCD:
- Start with the two numbers: 204 and 340.
- Divide 340 by 204:
\[ 340 \div 204 = 1 \quad \text{(quotient)} \]
\[ 340 - (204 \times 1) = 136 \quad \text{(remainder)} \]

5. **Continue the Process**:
Now, we take the previous divisor (204) and the remainder (136):
- Divide 204 by 136:
\[ 204 \div 136 = 1 \quad \text{(quotient)} \]
\[ 204 - (136 \times 1) = 68 \quad \text{(remainder)} \]

6. **Final Step**:
Now, divide 136 by the last remainder (68):
- Divide 136 by 68:
\[ 136 \div 68 = 2 \quad \text{(quotient)} \]
\[ 136 - (68 \times 2) = 0 \quad \text{(remainder)} \]
Since the remainder is now 0, we have found our GCD.

7. **Conclusion**:
The GCD of 204 and 340 is 68. Therefore, the greatest number that divides both 209 and 347 leaving the specified remainders is:
\[ \text{Required number} = 68 \]