Finding the Least 5-Digit Number Divisible by 14, 16, and 18
To find the least 5-digit number that is exactly divisible by 14, 16, and 18, we need to follow these steps:
Step 1: Find the Least Common Multiple (LCM)
- First, we need to determine the LCM of the numbers 14, 16, and 18.
- The prime factorization of the numbers is as follows:
- 14 = 2 × 7
- 16 = 2^4
- 18 = 2 × 3^2
- Now, we take the highest power of each prime number:
- 2: highest power is 2^4 (from 16)
- 3: highest power is 3^2 (from 18)
- 7: highest power is 7^1 (from 14)
- Therefore, the LCM is calculated as:
- LCM = 2^4 × 3^2 × 7 = 16 × 9 × 7 = 1008
Step 2: Find the Smallest 5-Digit Number
- The smallest 5-digit number is 10000.
- To find the smallest 5-digit number divisible by the LCM (1008), we need to divide 10000 by 1008:
- 10000 ÷ 1008 ≈ 9.92
- We round up to the next whole number, which is 10.
Step 3: Calculate the Result
- Now, multiply 10 by the LCM:
- 10 × 1008 = 10080
Conclusion
- The least 5-digit number that is exactly divisible by 14, 16, and 18 is:
- 10080