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

Answer

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

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

- The prime factorization of 12 is:
\[ 12 = 2^2 \times 3^1 \]
- The prime factorization of 15 is:
\[ 15 = 3^1 \times 5^1 \]

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

Thus, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]

### Step 2: Identify the greatest three-digit number
The greatest three-digit number is 999.

### Step 3: Check divisibility of 999 by 60
Now, we need to check if 999 is divisible by 60. We do this by performing the division:
\[ 999 \div 60 \]

Calculating this:
- \(60\) goes into \(999\) a total of \(16\) times (since \(60 \times 16 = 960\)).
- Now, we find the remainder:
\[ 999 - 960 = 39 \]

### Step 4: Adjust to find the greatest number divisible by 60
Since 999 leaves a remainder of 39 when divided by 60, we need to subtract this remainder from 999 to find the largest three-digit number that is divisible by 60:
\[ 999 - 39 = 960 \]

### Conclusion
Thus, the greatest three-digit number that is exactly divisible by both 12 and 15 is:
\[ \boxed{960} \]

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