To solve the problem, we need to find the smallest perfect square that is divisible by both 4 and 5, and then find its square roots.

Step 1: Find the Least Common Multiple (LCM)

The first step is to find the LCM of the numbers 4 and 5. This will give us the smallest number that is divisible by both 4 and 5.

• $4 = 2^2$
• $5 = 5^1$

The LCM of 4 and 5 is: $\text{LCM}(4, 5) = 2^2 \times 5^1 = 20$

Step 2: Find the Smallest Perfect Square

To make this LCM (which is 20) a perfect square, we need to multiply it by the smallest number that will result in a perfect square. A perfect square must have even powers for all prime factors.

• $20 = 2^2 \times 5^1$

Currently, $20$ has an odd power of 5. To make it a perfect square, we multiply by 5 to get: $20 \times 5 = 100 = 10^2$

Step 3: Square Roots

The square root of 100 is: $\sqrt{100} = 10$

Final Answer

The smallest perfect square divisible by 4 and 5 is 100, and its square roots are ±10.

Would you like more details or have any questions? Here are 5 related questions to consider:

1. What are the prime factors of 100?
2. How can you find the LCM of more than two numbers?
3. What is the difference between LCM and GCD?
4. How do you determine if a number is a perfect square?
5. What are the properties of perfect squares?

Tip: When finding a perfect square divisible by multiple numbers, start by finding the LCM and then adjust it to make it a perfect square.