Find the smallest square which is divisible by 2,3and5

To find the smallest square divisible by 2, 3, and 5, follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of 2, 3, and 5
The LCM of 2, 3, and 5 is their product:
\[
\text{LCM}(2, 3, 5) = 2 \times 3 \times 5 = 30
\]

### Step 2: Check for Square Divisibility
A perfect square has all prime factors raised to an even power. The prime factorization of 30 is:
\[
30 = 2^1 \times 3^1 \times 5^1
\]
To make 30 a perfect square, each prime exponent must be even. Thus, we need to multiply by an extra \(2^1\), \(3^1\), and \(5^1\) to raise each prime factor to an even power:
\[
30 \times 2 \times 3 \times 5 = 2^2 \times 3^2 \times 5^2 = 900
\]

### Step 3: Conclusion
The smallest square divisible by 2, 3, and 5 is \( \boxed{900} \).

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. What is the smallest square divisible by 2, 3, 5, and 7?
2. How do you find the smallest cube divisible by 2, 3, and 5?
3. Can you explain how to find the least common multiple (LCM) of numbers?
4. How can you determine if a number is a perfect square?
5. What is the smallest square divisible by 4, 9, and 25?

**Tip**: Always break down the prime factorization when dealing with divisibility problems, especially for perfect squares or cubes.

Find the smallest square which is divisible by 2, 3 and 5

Find the smallest square divisible by 2, 3, and 5 with a detailed step-by-step explanation. Learn about LCM, prime factorization, and the properties of perfect squares. This solution is suitable for high school students in Grades 9-11.

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