Find the least number of five digits that is exactly divisible by 16, 18, 24 and 30 .

Answer

To find the least number of five digits that is exactly divisible by 16, 18, 24, and 30, we will follow these steps:

### Step 1: Find the LCM of the numbers
To find a number that is divisible by 16, 18, 24, and 30, we first need to calculate the Least Common Multiple (LCM) of these numbers.

- **Prime factorization**:
- 16 = 2^4
- 18 = 2^1 × 3^2
- 24 = 2^3 × 3^1
- 30 = 2^1 × 3^1 × 5^1

- **Taking the highest powers of each prime**:
- For 2: max(4, 1, 3, 1) = 4 → 2^4
- For 3: max(0, 2, 1, 1) = 2 → 3^2
- For 5: max(0, 0, 0, 1) = 1 → 5^1

Thus, LCM = 2^4 × 3^2 × 5^1.

### Step 2: Calculate the LCM
Now we calculate the LCM:

- LCM = 16 × 9 × 5
- LCM = 144 × 5 = 720

### Step 3: Find the smallest 5-digit number
The smallest 5-digit number is 10,000. We need to find the smallest multiple of 720 that is greater than or equal to 10,000.

### Step 4: Divide to find the multiple
Now we divide 10,000 by 720 to find how many times 720 fits into 10,000:

10,000 ÷ 720 ≈ 13.8889

### Step 5: Round up to the next whole number
Since we need a whole number, we round up 13.8889 to the next whole number, which is 14.

### Step 6: Multiply to find the least 5-digit number
Now we multiply 720 by 14 to find the least 5-digit number:

720 × 14 = 10,080

### Conclusion
The least number of five digits that is exactly divisible by 16, 18, 24, and 30 is **10,080**.

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