Introduction to Linear Diophantine Equations

Linear Diophantine equations are mathematical equations that involve two or more variables and require integer solutions. These equations are fundamental in number theory and have practical applications in various fields, including cryptography and computer science. In this article, we will explore the process of finding positive integral solutions to the equation 2x - 5y 17, a common example in the study of these equations.

Understanding the Equation 2x - 5y 17

The equation we are analyzing is 2x - 5y 17. Our goal is to find all positive integer pairs (x, y) that satisfy this equation. To achieve this, we will follow a systematic approach, starting from basic transformations and conditions.

Step 1: Isolating y

First, we can isolate y in terms of x:

2x - 5y 17

5y 17 - 2x

y (17 - 2x) / 5

Step 2: Conditions for Positive Integral Solutions

For y to be a positive integer, the expression 17 - 2x must be non-negative and a multiple of 5. Let's explore these conditions:

Condition 1: Non-Negative Expression

To ensure that the expression 17 - 2x is non-negative:

17 - 2x ≥ 0

2x ≤ 17

x ≤ 8.5

Since x must be a positive integer, we have:

x ≤ 8

Condition 2: Multiple of 5

The expression 17 - 2x must be a multiple of 5:

17 - 2x ≡ 0 (mod 5)

Let's find the value of 17 mod 5:

17 mod 5 2

Thus, we need:

2 - 2x ≡ 0 (mod 5)

2x ≡ 2 (mod 5)

x ≡ 1 (mod 5)

This implies that x can be expressed as:

x 5k 1

where k is a non-negative integer.

Evaluating Possible Values for x

We need to ensure that x remains within the bounds established earlier, i.e., x ≤ 8. Let's evaluate possible values of k:

If k 0, then x 5*0 1 1 If k 1, then x 5*1 1 6 If k 2, then x 5*2 1 11 (exceeds 8, not valid)

Thus, the valid values for x are 1 and 6.

Finding Corresponding y Values

Now, we substitute these values back into the equation to find the corresponding values of y:

If x 1: 2*1 - 5y 17 2 - 5y 17 5y -15 y 3 One solution is (1, 3) If x 6: 2*6 - 5y 17 12 - 5y 17 5y -5 y 1 Another solution is (6, 1)

Conclusion: Positive Integral Solutions to 2x - 5y 17

The positive integral solutions to the equation 2x - 5y 17 are:

(1, 3) (6, 1)