Find the greatest 3-digit number which is divisible by 16,18 and 24.

Explanation

To find the greatest 3-digit number that is divisible by 16, 18, and 24, we will follow these steps:

### Step 1: Find the LCM of 16, 18, and 24
To find the least common multiple (LCM) of these numbers, we can use the prime factorization method.

- **Prime factorization of 16**:
- \(16 = 2^4\)

- **Prime factorization of 18**:
- \(18 = 2^1 \times 3^2\)

- **Prime factorization of 24**:
- \(24 = 2^3 \times 3^1\)

Now, we take the highest power of each prime number present in the factorizations:

- For \(2\): The highest power is \(2^4\) (from 16).
- For \(3\): The highest power is \(3^2\) (from 18).

Thus, the LCM is:
\[ LCM = 2^4 \times 3^2 = 16 \times 9 = 144 \]

### Step 2: Find the greatest 3-digit number divisible by 144
The greatest 3-digit number is 999. To find the largest multiple of 144 that is less than or equal to 999, we can divide 999 by 144 and take the integer part of the result.

\[ 999 \div 144 \approx 6.9375 \]

Taking the integer part, we get 6. Now, we multiply 144 by 6 to find the greatest multiple:

\[ 144 \times 6 = 864 \]

### Step 3: Verify that 864 is divisible by 16, 18, and 24
To ensure that 864 is divisible by all three numbers, we can perform the following checks:

- **Divisibility by 16**:
\[ 864 \div 16 = 54 \quad (\text{exact, so divisible}) \]

- **Divisibility by 18**:
\[ 864 \div 18 = 48 \quad (\text{exact, so divisible}) \]

- **Divisibility by 24**:
\[ 864 \div 24 = 36 \quad (\text{exact, so divisible}) \]

### Conclusion
The greatest 3-digit number that is divisible by 16, 18, and 24 is **864**.

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