To find the smallest square number divisible by each of the numbers 16, 18, 24, 28, and 30, we need to follow these steps:

1. Find the prime factorization of each number.

   • $16 = 2^4$
   • $18 = 2 \times 3^2$
   • $24 = 2^3 \times 3$
   • $28 = 2^2 \times 7$
   • $30 = 2 \times 3 \times 5$

2. Determine the least common multiple (LCM) of these numbers.

   The LCM is found by taking the highest power of each prime number that appears in the factorizations:

   • The highest power of $2$ is $2^4$ (from 16).
   • The highest power of $3$ is $3^2$ (from 18).
   • The highest power of $5$ is $5^1$ (from 30).
   • The highest power of $7$ is $7^1$ (from 28).

   So, the LCM is: $\text{LCM} = 2^4 \times 3^2 \times 5^1 \times 7^1$

3. Calculate the LCM:

   $2^4 = 16, \quad 3^2 = 9, \quad 5^1 = 5, \quad 7^1 = 7$ $\text{LCM} = 16 \times 9 \times 5 \times 7$

   Let's compute it step-by-step:

   $16 \times 9 = 144$ $144 \times 5 = 720$ $720 \times 7 = 5040$

   So, $\text{LCM} = 5040$.

4. Find the smallest square number divisible by 5040.

   To be a square number, each prime factor's power must be even. Let's adjust the LCM to make each exponent even:

   • $2^4$ is already even.
   • $3^2$ is already even.
   • $5^1$ is odd, so we need $5^2$.
   • $7^1$ is odd, so we need $7^2$.

   Thus, the smallest square number is: $2^4 \times 3^2 \times 5^2 \times 7^2$

   Calculating this: $2^4 = 16, \quad 3^2 = 9, \quad 5^2 = 25, \quad 7^2 = 49$ $16 \times 9 = 144$ $144 \times 25 = 3600$ $3600 \times 49 = 176400$

Therefore, the smallest square number that is divisible by each of the numbers 16, 18, 24, 28, and 30 is 176400.

Would you like more details or have any questions?

Here are five related questions for further exploration:

1. How do you determine the LCM of two or more numbers using their prime factorizations?
2. What is the significance of even exponents in the context of square numbers?
3. How can the method used here be applied to find the smallest cube number divisible by a set of numbers?
4. What is the role of prime factorization in number theory?
5. How does finding the LCM help in solving problems related to divisibility and multiples?

Tip: When working with large numbers, break down the problem into smaller, manageable parts, such as prime factorization, and then use these parts to build the solution step-by-step.