To find the smallest square number that is divisible by the numbers 4, 12, and 16, follow these steps:
1. Identify the Least Common Multiple (LCM):
- First, we need to determine the LCM of the numbers 4, 12, and 16.
- The LCM of a set of numbers is the smallest number that is evenly divisible by all of them.
2. Prime Factorization:
- Perform prime factorization for each number:
- 4 = 2^2
- 12 = 2^2 3
- 16 = 2^4
- To find the LCM, take the highest power of each prime factor appearing in the factorizations:
- The highest power of 2 is 2^4.
- The highest power of 3 is 3^1.
- So, the LCM is 2^4 3 = 16 * 3 = 48.
3. Determine the Smallest Square Number Divisible by the LCM:
- Now, we need to find the smallest square number that is divisible by 48.
- A square number is a number that can be expressed as the product of an integer with itself (e.g., 1, 4, 9, 16, ...).
- To find this number, we take the square root of 48, which is approximately 6.93.
- The smallest integer greater than 6.93 is 7.
- Finally, squaring 7 gives us 7^2 = 49.
Conclusion:
The smallest square number which is divisible by the numbers 4, 12, and 16 is 49.
