Introduction to Perfect Squares and Divisibility

Perfect squares are numbers that are the square of an integer. In other words, a perfect square is a number that can be expressed as the product of an integer with itself. For instance, 36 is a perfect square because it is 62. Understanding perfect squares and divisibility is crucial when dealing with mathematical problems, especially in the context of Least Common Multiples (LCM) and prime factorization. This article will explore how to find the least perfect square divisible by given numbers using LCM and prime factorization techniques, in a manner that adheres to Google's SEO standards.

Finding the Least Perfect Square Divisible by 3, 4, 5, 6, and 8

First, let's address the problem of finding the least perfect square divisible by 3, 4, 5, 6, and 8.

Step 1: Compute the LCM of 3, 4, 5, 6, and 8.

Prime factorization: 3 3 4 22 5 5 6 2 × 3 8 23

Step 2: Use the LCM of these numbers. The LCM is 23 × 3 × 5 120.

Step 3: To make 120 a perfect square, we need to make sure all the prime factors are raised to an even power. Here, 120 23 × 31 × 51.

Step 4: We need to multiply 120 by 22 (which is 4) to make all the prime factors even:

120 × 4 120 × 22 (23 × 22) × (3 × 5) 2? × 3 × 5 ≡ (22)2 × 3? × 5? 480 (not a perfect square yet)

To make it a perfect square, we need to multiply by 3 × 5 15.

120 × 15 1800

1800 2? × 32 × 52, which is a perfect square (302 900).

Additional Examples

Example 1: Finding the least perfect square divisible by 3 and 4.

LCM of 3 and 4 is 12. To make 12 a perfect square, we need to multiply it by 3 and 4 to make all prime factors even powers.

12 × 3 × 4 12 × 12 144 122.

Example 2: Finding the least perfect square divisible by 34, 5.

LCM of 34 and 5 is 170. To make 170 a perfect square, we need to multiply it by 2 × 3, 5, and 11.

170 × 2 × 3 × 5 × 11 170 × 330 56100 2702.

Example 3: Finding the least perfect square divisible by 3456810 and 11.

LCM of 3456810 and 11 is 1320. To make 1320 a perfect square, we need to multiply it by 3, 5, and 11.

1320 × 3 × 5 × 11 1320 × 165 217800 6602.

Example 4: Finding the least perfect square divisible by 4 and 7.

LCM of 4 and 7 is 28. Since 28 is not a perfect square, we need to multiply it by 7 to make all prime factors even.

28 × 7 196 142.

Conclusion

In conclusion, finding the least perfect square divisible by given numbers involves finding the LCM and then making all the prime factors even powers. This process involves understanding the prime factorization and the concept of LCM. By following this method, you can solve similar problems efficiently.

Keywords: Least Perfect Square, Divisibility, LCM, Prime Factors

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