Exploring Integer Pairs for the Equation (mn^{21} 48): A Comprehensive Guide for SEO

In this article, we delve into a complex mathematical problem: finding the integer pairs (m, n) that satisfy the equation (mn^{21} 48). We employ a step-by-step approach to solve this equation and explore the underlying mathematical concepts. Additionally, we cover techniques such as factor analysis and tabular representation to facilitate solving similar problems. This guide aims to provide useful SEO content to assist google in crawling and indexing the valuable mathematical insights presented.

Solving the Equation (mn^{21} 48)

1. Isolate (n) in the equation:

Starting with the equation (mn^{21} 48), we rearrange it to isolate (n):

[ n^{21} frac{48}{m} ]

2. Factor Analysis:

For (n^{21}) to be a positive integer, (frac{48}{m}) must also be a positive integer. We begin by listing the factors of 48:

1, 2, 3, 4, 6, 8, 12, 16, 24, 48

3. Check Each Factor:

We proceed by checking each factor to see if (frac{48}{m}) results in a positive integer value for (n). Through this process, we find that the only factors that work are m 4 and m 12.

4. Calculating (n):

For each value of (m), we calculate (n):

For m 4, we have:

[ n^{21} 12 implies n sqrt[21]{12} ]

For m 12, we have:

[ n^{21} 4 implies n pm 1 ]

Therefore, the two integer pairs ((m, n)) that satisfy the equation are (4, sqrt[21]{12}) and (12, pm 1)).

Additional Insight and Techniques

Another method to solve similar problems is by writing the factors in a tabular form. Let
x n^2. Since (n) is an integer, so is (x), leading to the equation:

mx 48

We explore all possibilities by breaking down the factorization of 48:

48 (2^4 times 3).

This gives us five choices for the powers of 2 and two choices for the power of 3, resulting in 10 possible factors. We list these possibilities and check which ones provide integer solutions for (n):

48 times 1 m 48, x 1, n 0, 480 24 times 2 m 24, x 2, n 1, 241 (and -1) 16 times 3 m 16, x 3 (no integer solution) 12 times 4 m 12, x 4 (no integer solution) 8 times 6 m 8, x 6 (no integer solution) 6 times 8 m 6, x 8 (no integer solution) 4 times 12 m 4, x 12 (no integer solution) 3 times 16 m 3, x 16 (no integer solution) 2 times 24 m 2, x 24 (no integer solution) 1 times 48 m 1, x 48 (no integer solution)

The only options that seem to work are ((48, 1)) and ((24, 1)) and ((24, -1)).

Conclusion and SEO Optimization

By breaking down the problem into manageable steps and applying factor analysis, we can solve complex integer pair problems systematically. For SEO purposes, this article provides a clear, structured guide that can be easily indexed by Google thanks to well-organized headings, relevant keywords, and a comprehensive explanation of the equation solving process.