Find the largest number that divides 280 and 1248 leaving remainders 4 and 6 respectively.

Answer

To solve the problem of finding the largest number that divides 280 and 1248, leaving remainders of 4 and 6 respectively, we can follow these steps:

### Step 1: Adjust the Dividends
We start by adjusting the dividends by subtracting the respective remainders from each number.

- For 280, subtract 4:
\[ 280 - 4 = 276 \]
- For 1248, subtract 6:
\[ 1248 - 6 = 1242 \]

### Step 2: Find the HCF (Highest Common Factor)
Next, we need to find the HCF of the adjusted numbers 276 and 1242. We can use the prime factorization method or the Euclidean algorithm. Here, we will use the Euclidean algorithm.

1. Divide 1242 by 276 and find the remainder:
\[ 1242 \div 276 = 4 \quad \text{(quotient)} \]
\[ 1242 - (276 \times 4) = 1242 - 1104 = 138 \quad \text{(remainder)} \]

2. Now, apply the same process to 276 and 138:
\[ 276 \div 138 = 2 \quad \text{(quotient)} \]
\[ 276 - (138 \times 2) = 276 - 276 = 0 \quad \text{(remainder)} \]

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

### Step 3: Conclusion
The HCF of 276 and 1242 is 138. Therefore, the largest number that divides both 280 and 1248, leaving remainders of 4 and 6 respectively, is:
\[ \text{Largest number} = 138 \]