Finding the Least Positive Integer K for a Perfect Cube

When dealing with the mathematical concept of a perfect cube, we often need to find the smallest positive integer K such that a given product is a perfect cube. This article will guide you through the process using two different examples, involving the numbers 2000, 2001, and 2009, and demonstrate the techniques used to solve such problems.

Introduction

A perfect cube is a number that can be expressed as the cube of an integer. For example, 27 is a perfect cube because it can be written as (3^3).

Example 1: Finding K for 2000, 2009, and K

Step-by-Step Solution

Factor the numbers 2000 and 2009 into their prime factors:

2000 (2^4 times 5^3) 2009 (7^2 times 41^1)

Combine the factors of 2000 and 2009:

2000 × 2009 (2^4 times 5^3 times 7^2 times 41^1)

Analyze the exponents of the prime factors to ensure they are multiples of 3:

For (2^4), the exponent 4 needs to be adjusted to the nearest multiple of 3, which is 6, so we need (2^2). For (5^3), the exponent 3 is already a multiple of 3, so no additional factors are needed. For (7^2), the exponent 2 needs to be adjusted to 3, so we need (7^1). For (41^1), the exponent 1 needs to be adjusted to 3, so we need (41^2).

Calculate the value of K:

22 4 71 7 412 1681 K 22 × 7 × 412 4 × 7 × 1681 28 × 1681 47068

Thus, the least positive integer K such that 2000 times 2009 times K is a perfect cube is 47068.

Example 2: Finding K for 2000 and 2001

In another example, we need to find the smallest positive integer K such that the product of 2000, 2001, and K is a perfect cube.

Prime Factorization and Analysis

2000 (2^4 times 5^3) 2001 (3 times 23 times 29) Product modulo 5: 2000 × 2001 (5^3 times 2^4 times 3 times 23 times 29) Only the prime factor 5 appears as a cube. The factor 2 needs an extra '2', the factors 3, 23, and 29 are square-free.

Step-by-Step Solution

To make the product a perfect cube, we need to square the factors 2, 3, 23, and 29.

The least number K that must be multiplied to achieve this is:

22 × 32 × 232 × 292 16016004

When we multiply 4002000 by 16016004, we get 64096048008000, which is a perfect cube since (sqrt[3]{64096048008000} 40020), and 40020 is an integer.

Hence, the least value of K is 16016004.

Conclusion

Through the examples detailed above, we have successfully demonstrated the process of finding the least positive integer K such that a given product is a perfect cube. This involves factorizing the given numbers, analyzing the exponents, and determining the necessary adjustments to make the product a perfect cube.