Solving for Integer Pairs with Specific Sum and Product Conditions

Objective: Find the possible integer pairs that satisfy the conditions of having a specific sum and a product that is at most a certain value.

Introduction

This article explores the problem of finding pairs of positive integers x and y that satisfy the following conditions:

Their product is at most 24 (i.e., xy leq 24). Their sum is exactly 10 (i.e., x y 10).

Step-by-Step Solution

Let us start by expressing y in terms of x using their sum condition:

y 10 - x.

Substituting y into the Product Inequality

Next, substitute y 10 - x into the product inequality:

xy leq 24 becomes x(10 - x) leq 24 or 1 - x^2 leq 24.

With some rearranging, we get:

x^2 - 1 24 geq 0.

Solving the Quadratic Equation

To find the roots of the quadratic equation x^2 - 1 24 0, we use the quadratic formula:

x frac{-b pm sqrt{b^2 - 4ac}}{2a} where a 1, b -10, and c 24.

Substituting these values into the formula, we get:

x frac{10 pm sqrt{(-10)^2 - 4 cdot 1 cdot 24}}{2 cdot 1} frac{10 pm sqrt{100 - 96}}{2} frac{10 pm 2}{2}.

This yields the roots:

x 6 and x 4.

Factoring the Quadratic Inequality

We can now factor the quadratic inequality:

(x - 4)(x - 6) geq 0.

Intervals Based on Roots

The inequality holds for the following intervals:

x leq 4 6 leq x leq 10 (but since x y 10, we need to ensure x leq 10) x geq 6

Checking Positive Integer Pairs

We now check the integer values for x within these intervals to find pairs that satisfy both conditions:

x leq 4 x 1, y 9, product 9 x 2, y 8, product 16 x 3, y 7, product 21 x 4, y 6, product 24 x geq 6 x 6, y 4, product 24 x 7, y 3, product 21 x 8, y 2, product 16 x 9, y 1, product 9

Combining the pairs, we get:

(1, 9) (2, 8) (3, 7) (4, 6) (6, 4) (7, 3) (8, 2) (9, 1)

These pairs satisfy both conditions: the sum is 10 and the product is at most 24.

Verification Using J Programming Language

The J programming language can be used to verify the pairs as follows:

{n~10/│1 9│2 8│3 7│4 6│

This confirms that the 4 integer pairs that meet all the criteria are (1, 9), (2, 8), (3, 7), and (4, 6).

Conclusion

The possible pairs of positive integers that satisfy the conditions of having a specific sum and a product that is at most 24 are: