Introduction

Today, we will explore a challenging problem in number theory and algebra. We are tasked with finding all positive integer pairs (x, y) such that the product of their square roots multiplied by the square root of 2016 equals 12 times the square root of 14. This problem requires a deep understanding of algebraic integers and factorization properties.

Problem Formulation

The given problem can be stated as follows:

#8730;x#8730;y#8730;2016 12#8730;14

This equation can be rewritten as:

#8730;xy#8730;2016 12#8730;14

Squaring both sides, we obtain:

xy2#8730;xy 2016

2016 - xy 2#8730;xy

Since the left-hand side is an integer, the right-hand side must also be an integer.

Solution Methodology

We will proceed using the substitutions:

x 14a^2, y 14b^2

Substituting these into the equation, we get:

#8730;14a^2 #8730;14b^2 12#8730;14

14ab 12

ab 12 / 14 * m 12

ab 12

To find all possible pairs (a, b) where ab 12, we need to consider the divisors of 12. The possible values for a and b are:

(a, b) (1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1)

To each pair (a, b), we can assign a corresponding (x, y) pair as follows:

x 14a^2, y 14(12 - a^2)

For each value of a from 1 to 11, we get the corresponding (x, y) pairs.

Example Calculation

Let's take an example where a 2:

x 14(2^2) 14(4) 56

y 14(12 - 2^2) 14(8) 112

Therefore, one of the pairs (x, y) that satisfies the given condition is (56, 112).

Similarly, we can calculate for other values of a from 1 to 11, resulting in 11 distinct pairs of (x, y).

Let's summarize the pairs for each value of a:

ax 14a^2y 14(12 - a^2) 156112 211256 3168224 4224168 530888 639288 751814 867214 986814 10109214 11134414

Conclusion

In conclusion, the task of finding all positive integer pairs (x, y) that satisfy the given condition can be efficiently solved using algebraic substitutions and factorization properties. The pairs (x, y) can be systematically determined by considering the divisor pairs of 12.

Keywords

- Positive Integers

- Algebraic Integers

- Integer Factorization

- Square Terms