Question

Solve \(1096 x+1=3 y\) using Brahmagupta's method. Given a solution to this equation (with "additive" 1 ), it is easy to find solutions to equations with other additives by simply multiplying. For example, solve \(1096 x+10=3 y\).

Short answer

Answer: The positive integer solution for the equation 1096x + 10 = 3y is x = 3670 and y = 1333190.

Step-by-step solution

Find the greatest common divisor (GCD) of the coefficients

Calculate the GCD of the coefficients 1096 and 3. In this case, their GCD is 1, because they are both prime numbers and have no other common factors.

Apply Brahmagupta's method

First, we need to divide the equation 1096x + 1 = 3y by the GCD (which is 1), to confirm that it is solvable. After dividing the equation by the GCD, we are left with the same equation. Now, according to Brahmagupta's method, we let 1 as the result of 1096 * x + 3 * (-y). We can rewrite the equation 1096x + 1 = 3y as 1096x - 3y = -1. Next, we will find a solution for the modified equation by using the equation ax + by = gcd(a, b). A solution for this would be x = 1 and y = -365. So, we obtain the equation as: \(1096(1) + 3(-365) = -1\)

Find a solution for the original equation

Now that we have a solution for the modified equation, we need to find a solution for the original equation. Since we are looking for positive integer solutions for x and y, we need to adjust our previous solution to obtain positive integers. We see that in the modified equation, x = 1 and y = -365, the y-value is negative. To find positive integer solutions, we'll use the formula x' = x + bt and y' = y + at, where x' and y' are the new values of x and y, and t is an integer which we will choose to get positive integer solutions. For t = 122, we have x' = 1 + 3(122) = 367 and y' = -365 + 1096(122) = 133319. So, the positive integer solution for the original equation would be x = 367 and y = 133319.

Solve the modified equation with a different additive

Now, we need to find a solution for the equation with a different additive: \(1096x + 10 = 3y\). We can see that the new additive is 10 times the original additive, which was 1. Therefore, we can multiply the original solution (x, y) by 10 to obtain the solution for the new equation. For x' = 367 and y' = 133319, the new equation becomes: \(1096(3670) + 10 = 3(1333190)\) So, the solution for the modified equation with a different additive is x = 3670 and y = 1333190.

Key concepts

Diophantine Equations

Diophantine equations are a fascinating area of mathematics that deal exclusively with integer solutions. These equations are named after the ancient Greek mathematician Diophantus, and they form the basis for many number theory problems.

A Diophantine equation usually involves variables and some specific set of conditions. Typically, the coefficients and solutions are required to be integers. In our exercise, the focus is on a linear Diophantine equation of the form \( ax + by = c \). Here, the challenge is to find integer values for \(x\) and \(y\) that satisfy the equation.

One significant second step in solving Diophantine equations, like in this exercise, involves simplifying the equation by the greatest common divisor and leveraging Brahmagupta's method to find particular solutions. Understanding this framework is essential for approaching more advanced integer problems.

Some properties include:

• If the greatest common divisor (GCD) of \(a\) and \(b\) divides \(c\), then the equation has integer solutions.
• It is a common step to modify equations to find a particular type of solution with specific characteristics.

Greatest Common Divisor

The greatest common divisor (GCD) is a fundamental concept that assists in simplifying equations, particularly Diophantine equations. The GCD of two numbers is the largest integer that divides both of them without leaving a remainder. For example, in our problem, the coefficients are 1096 and 3. Calculating their GCD helps determine solvability.

In this exercise, the GCD is 1, meaning 1096 and 3 are co-prime. This simplifies the idea that there are integer solutions to the given equation as long as the constant term is also divisible by the GCD. Here the constant term is 1, so an integer solution is possible.

How to calculate the GCD:

• In simpler cases, you can use a prime factorization.
• The Euclidean algorithm is a more systematic approach, especially useful for larger numbers.

Knowing the GCD provides a powerful tool not just for verifying the existence of solutions but also for simplifying the problem into a more manageable form by dividing through by the GCD.

Positive Integer Solutions

Finding positive integer solutions is often paramount because many practical problems require positive rather than negative numbers. After establishing a solution to a Diophantine equation, experts adjust it to ensure both \(x\) and \(y\) are positive integers.

In this exercise, our initial solution gives \(x = 1\) and \(y = -365\). However, we seek positive integers for these variables. To adjust these values into their positive counterparts, we use a technique involving a parameter \(t\) as follows:

Given adjustments:

• Use the formula \(x' = x + bt\) and \(y' = y + at\).
• Choose a suitable integer \(t\) until both \(x'\) and \(y'\) become positive.

In this exercise, the value \(t = 122\) transforms \(x\) and \(y\) into 367 and 133319, respectively. This creative adjustment process ensures solutions meet specific requirements of positivity and integers, which are often necessary for real-world scenarios.