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

1. Prime Factorization: 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. Least Common Multiple (LCM): Find the least common multiple (LCM) of the numbers by taking the highest powers of all prime factors appearing in the factorizations.

   • For $2$: The highest power is $2^4$ from 16.
   • For $3$: The highest power is $3^2$ from 18.
   • For $5$: The highest power is $5^1$ from 30.
   • For $7$: The highest power is $7^1$ from 28.

   Therefore, the LCM is:

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

3. Calculate the LCM: $\text{LCM} = 16 \times 9 \times 5 \times 7 = 144 \times 35 = 5040$

4. Smallest Square Number: To find the smallest square number divisible by this LCM, every prime factor in the factorization of the LCM must be raised to an even power.

   • For $2$: It is already $2^4$, an even power.
   • For $3$: It is $3^2$, an even power.
   • For $5$: It is $5^1$, we need $5^2$.
   • For $7$: It is $7^1$, we need $7^2$.

   Therefore, the smallest square number divisible by all these numbers is: $(2^4 \times 3^2 \times 5^2 \times 7^2) = 16 \times 9 \times 25 \times 49$

5. Calculate the Square Number: $\text{Smallest square number} = 16 \times 9 \times 25 \times 49 = 144 \times 1225 = 176400$

Final Answer: The smallest square number that is divisible by 16, 18, 24, 28, and 30 is 176400.

Do you need more details or have any questions about this solution?

Here are 5 related questions for further practice:

1. What is the smallest square number divisible by 12, 15, and 20?
2. How do you find the LCM of three numbers using their prime factorizations?
3. What is the smallest square number divisible by 8, 14, 21, and 35?
4. How do you calculate the square root of a large number like 176400?
5. What are the steps to determine if a given number is a perfect square?

Tip: Always ensure each prime factor in the LCM has an even exponent to find the smallest square number.