Understanding the Least Real Number k to Make a Square from 315k

When dealing with equations involving square numbers, a common challenge is determining the smallest value of k that makes 315k a perfect square. This article provides a detailed explanation on how to solve such a problem, presenting the concept of prime factorization and why it is a key tool in solving these types of questions. Additionally, we'll explore the significance of having even exponents in the prime factorization of a quadratic number.

Prime Factorization and Square Numbers

A square number is a number that can be expressed as the product of an integer with itself. For instance, 16 is a square number because 16 42. A number is a square if and only if all the exponents in its prime factorization are even. This property is crucial for solving our problem.

Prime Factorization of 315

To determine the smallest k that makes 315k a square, we first find the prime factorization of 315. The prime factorization of 315 is:

315 32 × 5 × 7

Identifying Necessary Multiplication

In the prime factorization of 315, the exponents of 3, 5, and 7 are 2, 1, and 1, respectively. To make 315k a perfect square, we need to ensure that all the exponents in the prime factorization of the resulting number are even. Currently, the exponents of 5 and 7 are 1, which is odd. To make them even, we need to multiply by 5 and 7, respectively. Therefore, the smallest k that makes 315k a square is the product of 5 and 7.

Calculation of k

[text{k} 5 times 7 35]

Calculating the Square

Multiplying 315 by 35, we get:

315 × 35 11025

And 11025 is a perfect square because:

[sqrt{11025} 105]

Thus, 11025 1052.

Conclusion

The least real number k that makes 315k a square number is 35. This is because multiplying 315 by 35 makes all the exponents in the prime factorization even.

Further Exploration

The concept of square numbers and prime factorization is not only limited to the number 315. This same method can be applied to any number to find the smallest k that makes a product a perfect square. Understanding this principle can be incredibly useful in various areas of mathematics, such as number theory and algebra.

References

For further reading and exploration, consider the following relevant texts: