Find the greatest number of 3 digits which is exactly divisible by 12 and 15.

Answer

To find the greatest 3-digit number that is exactly divisible by both 12 and 15, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of 12 and 15
To find the LCM, we can use the prime factorization method.

- **Prime factorization of 12**:
- 12 = 2^2 × 3^1
- **Prime factorization of 15**:
- 15 = 3^1 × 5^1

Now, take the highest power of each prime number:
- For 2: 2^2 (from 12)
- For 3: 3^1 (common in both)
- For 5: 5^1 (from 15)

Now, we can calculate the LCM:
\[ \text{LCM} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]

### Step 2: Identify the Greatest 3-Digit Number
The greatest 3-digit number is 999.

### Step 3: Find the Largest Multiple of 60 Less Than or Equal to 999
To find the largest multiple of 60 that is less than or equal to 999, we can divide 999 by 60 and take the floor of the result.

\[ 999 \div 60 \approx 16.65 \]
Taking the floor gives us 16.

### Step 4: Calculate the Largest Multiple of 60
Now, multiply 60 by 16 to find the largest 3-digit number that is a multiple of 60.

\[ 60 \times 16 = 960 \]

### Conclusion
The greatest 3-digit number that is exactly divisible by both 12 and 15 is **960**.

---