Find the largest number which divides 307 and 330 and leaves remainder 3 and 7 respectively?

Options

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

Answer

To find the largest number that divides 307 and 330 leaving remainders of 3 and 7 respectively, we can follow these steps:

### Step 1: Adjust the Numbers
First, we need to adjust the numbers 307 and 330 by subtracting their respective remainders.

- For 307, subtract the remainder 3:
\[ 307 - 3 = 304 \]
- For 330, subtract the remainder 7:
\[ 330 - 7 = 323 \]

### Step 2: Find the HCF (Highest Common Factor)
Next, we need to find the HCF of the adjusted numbers 304 and 323. We can use the Euclidean algorithm for this.

1. Divide 323 by 304:
\[ 323 \div 304 = 1 \quad \text{(quotient)} \quad \text{remainder} = 323 - 304 = 19 \]
2. Now, divide 304 by the remainder 19:
\[ 304 \div 19 = 16 \quad \text{(quotient)} \quad \text{remainder} = 304 - (19 \times 16) = 304 - 304 = 0 \]

Since the remainder is now 0, the last non-zero remainder is the HCF.

### Step 3: Conclusion
The HCF of 304 and 323 is 19. Therefore, the largest number that divides 307 and 330 leaving remainders of 3 and 7 respectively is:
\[ \text{Answer} = 19 \]

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