Find the greatest number which divides 219 and 486 leaving remainder 3 and 2 respectively.

Answer

To solve the problem of finding the greatest number that divides 219 and 486, leaving remainders of 3 and 2 respectively, we can follow these steps:

### Step 1: Adjust the given numbers
We need to subtract the remainders from the given numbers to find the actual numbers that the greatest divisor will divide.

- For 219, subtract 3:
\[ 219 - 3 = 216 \]
- For 486, subtract 2:
\[ 486 - 2 = 484 \]

### Step 2: Find the HCF of the adjusted numbers
Now we need to find the highest common factor (HCF) of the two numbers we obtained: 216 and 484.

### Step 3: Factorize the numbers
Let's factorize both numbers.

- **Factorization of 216:**
\[ 216 = 2^3 \times 3^3 \]
(This means \(216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3\))

- **Factorization of 484:**
\[ 484 = 2^2 \times 11^2 \]
(This means \(484 = 2 \times 2 \times 11 \times 11\))

### Step 4: Identify common factors
Now, we identify the common factors from the factorizations:
- The common prime factor is \(2\).
- The lowest power of \(2\) in both factorizations is \(2^2\).

### Step 5: Calculate the HCF
Thus, the HCF of 216 and 484 is:
\[ HCF = 2^2 = 4 \]

### Conclusion
The greatest number that divides 219 and 486 leaving remainders of 3 and 2 respectively is:
\[ \text{Greatest number} = 4 \]

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