Find the largest number which divides 320 and 458 leaving remainders 5 and 8 respectively.

Answer

To find the largest number that divides 320 and 458 leaving remainders 5 and 8 respectively, we can follow these steps:

### Step 1: Subtract the remainders from the respective numbers
- For 320, subtract the remainder 5:
\[ 320 - 5 = 315 \]
- For 458, subtract the remainder 8:
\[ 458 - 8 = 450 \]

### Step 2: Find the HCF (Highest Common Factor) of the two results (315 and 450)
To find the HCF, we can use the prime factorization method.

#### 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:
\[ 315 = 3^2 \times 5^1 \times 7^1 \]

#### 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:
\[ 450 = 2^1 \times 3^2 \times 5^2 \]

### Step 3: Identify the common factors
The common prime factors between 315 and 450 are:
- For \(3\): The minimum power is \(3^2\)
- For \(5\): The minimum power is \(5^1\)

### Step 4: Calculate the HCF
Now we can calculate the HCF:
\[ HCF = 3^2 \times 5^1 = 9 \times 5 = 45 \]

### Conclusion
The largest number that divides 320 and 458 leaving remainders 5 and 8 respectively is:
\[ \boxed{45} \]