Find the largest number which divides 320 and 457 , leaving remainders 5 and 7 respectively .

Explanation

To find the largest number that divides 320 and 457, leaving remainders of 5 and 7 respectively, we can follow these steps:

### Step 1: Adjust the numbers by subtracting the remainders
We need to adjust the given numbers by subtracting their respective remainders.

- For 320, subtract 5:
\[ 320 - 5 = 315 \]

- For 457, subtract 7:
\[ 457 - 7 = 450 \]

### Step 2: Find the HCF (Highest Common Factor) of the adjusted numbers
Now, we need to find the HCF of 315 and 450.

#### Finding the prime factorization of 315:
1. Divide by 3:
\[ 315 \div 3 = 105 \]
2. Divide by 3 again:
\[ 105 \div 3 = 35 \]
3. Divide by 5:
\[ 35 \div 5 = 7 \]
4. 7 is a prime number.

So, the prime factorization of 315 is:
\[ 315 = 3^2 \times 5 \times 7 \]

#### Finding the prime factorization of 450:
1. Divide by 2:
\[ 450 \div 2 = 225 \]
2. Divide by 3:
\[ 225 \div 3 = 75 \]
3. Divide by 3 again:
\[ 75 \div 3 = 25 \]
4. Divide by 5:
\[ 25 \div 5 = 5 \]
5. 5 is a prime number.

So, the prime factorization of 450 is:
\[ 450 = 2 \times 3^2 \times 5^2 \]

### Step 3: Identify the common factors
Now, we identify the common factors from the prime factorizations:
- For 3: The minimum power is \(3^2\) (common in both).
- For 5: The minimum power is \(5^1\) (common in both).

### Step 4: Calculate the HCF
The HCF is obtained by multiplying the lowest powers of the common prime factors:
\[ HCF = 3^2 \times 5^1 = 9 \times 5 = 45 \]

### Conclusion
The largest number that divides both 320 and 457, leaving remainders of 5 and 7 respectively, is:
\[ \boxed{45} \]