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

Answer

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

### Step 1: Adjust the numbers by subtracting the remainders
First, we need to adjust the numbers 280 and 1248 by subtracting their respective remainders.

- For 280, subtract the remainder 4:
\[ 280 - 4 = 276 \]

- For 1248, subtract the remainder 6:
\[ 1248 - 6 = 1242 \]

### Step 2: Find the greatest common divisor (GCD)
Next, we need to find the greatest common divisor (GCD) of the adjusted numbers 276 and 1242. We can use the Euclidean algorithm to find the GCD.

1. Divide 1242 by 276 and find the remainder:
\[ 1242 \div 276 \approx 4.5 \quad \text{(take the integer part, which is 4)} \]
\[ 1242 - (276 \times 4) = 1242 - 1104 = 138 \]

2. Now, apply the Euclidean algorithm again with 276 and 138:
\[ 276 \div 138 = 2 \quad \text{(the integer part)} \]
\[ 276 - (138 \times 2) = 276 - 276 = 0 \]

Since the remainder is now 0, the last non-zero remainder is the GCD. Therefore, the GCD of 276 and 1242 is 138.

### Step 3: Conclusion
The largest number that divides both 280 and 1248 leaving remainders 4 and 6 respectively is:
\[ \boxed{138} \]