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

Answer

To solve the problem of finding 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 given numbers
We start by subtracting the remainders from the respective numbers:
- For 320 with a remainder of 5:
\[ 320 - 5 = 315 \]
- For 458 with a remainder of 8:
\[ 458 - 8 = 450 \]

### Step 2: Find the HCF (Highest Common Factor) of the two results
Now, we need to find the HCF of 315 and 450. We can do this using the prime factorization method or the division method.

#### Prime Factorization Method:
1. **Factor 315**:
- 315 is divisible by 3:
\[ 315 \div 3 = 105 \]
- 105 is divisible by 3:
\[ 105 \div 3 = 35 \]
- 35 is divisible by 5:
\[ 35 \div 5 = 7 \]
- 7 is a prime number.
- Therefore, the prime factorization of 315 is:
\[ 315 = 3^2 \times 5^1 \times 7^1 \]

2. **Factor 450**:
- 450 is divisible by 2:
\[ 450 \div 2 = 225 \]
- 225 is divisible by 3:
\[ 225 \div 3 = 75 \]
- 75 is divisible by 3:
\[ 75 \div 3 = 25 \]
- 25 is divisible by 5:
\[ 25 \div 5 = 5 \]
- 5 is a prime number.
- Therefore, the prime factorization of 450 is:
\[ 450 = 2^1 \times 3^2 \times 5^2 \]

### Step 3: Determine the HCF
To find the HCF, we take the lowest power of all prime factors present in both numbers:
- For 3: minimum power is \(3^2\)
- For 5: minimum power is \(5^1\)
- 7 is not common in both.

Thus, the HCF is:
\[ HCF = 3^2 \times 5^1 = 9 \times 5 = 45 \]

### Conclusion
The largest number that divides both 320 and 458 leaving remainders 5 and 8 respectively is:
\[ \text{Largest Number} = 45 \]

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