What is the greatest number that will divide 307 and 330 leaving remainder 3 and 7 respectively ?

Options

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

Answer

To solve the problem of finding the greatest number that will divide 307 and 330 leaving remainders of 3 and 7 respectively, we can follow these steps:

### Step 1: Adjust the numbers for the remainders
To find the number \( N \) that divides both 307 and 330 with the specified remainders, we first adjust the numbers by subtracting the remainders from them.

- 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.

### Step 3: Factorize both numbers
Let's factorize both numbers to find their HCF.

**For 304:**
- Divide by 2:
\[ 304 \div 2 = 152 \]
- Divide by 2 again:
\[ 152 \div 2 = 76 \]
- Divide by 2 again:
\[ 76 \div 2 = 38 \]
- Divide by 2 again:
\[ 38 \div 2 = 19 \]
- 19 is a prime number.

Thus, the prime factorization of 304 is:
\[ 304 = 2^4 \times 19 \]

**For 323:**
- Check divisibility by 19:
\[ 323 \div 19 = 17 \]
- Both 19 and 17 are prime numbers.

Thus, the prime factorization of 323 is:
\[ 323 = 19 \times 17 \]

### Step 4: Identify the common factors
Now, we can identify the common factors in the factorizations of 304 and 323.

- The common factor is 19.

### Step 5: Conclusion
The greatest number \( N \) that will divide both 307 and 330 leaving remainders of 3 and 7 respectively is:
\[ \text{Greatest number} = 19 \]

### Final Answer
The greatest number is **19**.

---