Find the greatest number of five digits exactly divisible by 9, 12, 15, 18 and 24.

Answer

To find the greatest five-digit number that is exactly divisible by 9, 12, 15, 18, and 24, we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
First, we need to find the LCM of the numbers 9, 12, 15, 18, and 24.

1. **Prime Factorization:**
- 9 = 3²
- 12 = 2² × 3
- 15 = 3 × 5
- 18 = 2 × 3²
- 24 = 2³ × 3

2. **Identify the highest powers of each prime:**
- For 2: highest power is 2³ (from 24)
- For 3: highest power is 3² (from 9 and 18)
- For 5: highest power is 5¹ (from 15)

3. **Calculate the LCM:**
\[ \text{LCM} = 2³ × 3² × 5¹ = 8 × 9 × 5 \]
\[ = 72 × 5 = 360 \]

### Step 2: Find the Greatest Five-Digit Number
The greatest five-digit number is 99999.

### Step 3: Determine the Largest Multiple of 360
Now, we need to find the largest multiple of 360 that is less than or equal to 99999.

1. **Divide 99999 by 360:**
\[ 99999 ÷ 360 \approx 277.775 \]
The largest integer less than or equal to 277.775 is 277.

2. **Multiply 277 by 360:**
\[ 277 × 360 = 99960 \]

### Step 4: Conclusion
Thus, the greatest five-digit number that is exactly divisible by 9, 12, 15, 18, and 24 is **99960**.

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