What is the side length of the smallest square design that could be formed from  tiles measuring 6 cm by 15 cm ? Assume the tiles cannot be cut.

To find the side length of the smallest square that can be formed using tiles measuring 6 cm by 15 cm without cutting the tiles, we need to determine the least common multiple (LCM) of the two dimensions: 6 cm and 15 cm.

This is because the side of the square must be a multiple of both 6 cm and 15 cm to ensure the tiles fit perfectly.

### Step-by-Step Calculation:

1. **Prime factorization of 6**:
   \[
   6 = 2 \times 3
   \]
2. **Prime factorization of 15**:
   \[
   15 = 3 \times 5
   \]
3. **LCM of 6 and 15**:  
   The LCM is found by taking the highest powers of all prime factors present:
   \[
   \text{LCM}(6, 15) = 2 \times 3 \times 5 = 30
   \]

Thus, the side length of the smallest square that can be formed is 30 cm.

### Conclusion:
The smallest square has a side length of **30 cm**.

Would you like further clarification or details on any part of this solution?

Here are 5 related questions for you to consider:
1. How many tiles would be needed to cover the 30 cm square?
2. How do you find the greatest common divisor (GCD) of two numbers?
3. What is the difference between LCM and GCD?
4. How can the concept of tiling be extended to three dimensions?
5. Can a rectangular shape other than a square be formed with the same tiles?

**Tip:** When solving problems involving fitting shapes, always check for the LCM or GCD to find the smallest or largest fitting dimensions.

What is the side length of the smallest square design that could be formed from tiles measuring 6 cm by 15 cm? Assume the tiles cannot be cut.

Discover how to find the side length of the smallest square that can be formed using 6 cm by 15 cm tiles. This step-by-step guide uses prime factorization to determine the LCM, providing a detailed explanation suitable for students in Grades 6-8. Learn how LCM helps in tiling problems and geometry.

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