## How do you find the general solution of a linear Diophantine equation?

For example,

- Input: 25x + 10y = 15.
- Output: General Solution of the given equation is. x = 3 + 2k for any integer m. y = -6 – 5k for any integer m.
- Input: 21x + 14y = 35.
- Output: General Solution of the given equation is. x = 5 + 2k for any integer m. y = -5 – 3k for any integer m.

**How do you solve linear Diophantine?**

Solve the linear Diophantine Equation 20x+16y=500,x,y∈Z+.

- Solution.
- Step 1: gcd(20,16)=4.
- Step 2: A solution is 4125=20(1)(125)+16(−1)(125).
- Step 3: Let u = x – 125 and v = y + 125.
- Step 4: In general, the solution to ax + by = 0 is x=bdk and y=-adk, kZ \ {0}, d=gcd(a,b).
- Step 5: Replace u and v.

**How do you find the number of solutions to a Diophantine equation?**

For any n≥0, the number of solutions of z=n is P1(n)=1. For any n≥0, the number of solutions of y+z=n is P2(n)=P1(n)+P1(n−1)+…P1(0)=(n+1)∗1=n+1.

### How do you find ax by gcd AB?

If we can solve ax+by=gcd(a,b) then we can solve for any ax+by=d where d is a multiple of the greatest common divisor (how?). To do this, write ax+by=a’ gcd(a,b)x+b’ gcd(a,b)y=gcd(a,b) and divide through by gcd(a,b), giving a’x+b’y=1 where gcd(a’,b’)=1.

**How do you know if a Diophantine equation has a solution?**

Let a, b and c be integers with a≠0 and b≠0, and let d=gcd(a,b). If d does not divide c, then the linear Diophantine equation ax+by=c has no solution.

**When Diophantine equation has no solution?**

Applied to the simplest Diophantine equation, ax + by = c, where a, b, and c are nonzero integers, these methods show that the equation has either no solutions or infinitely many, according to whether the greatest common divisor (GCD) of a and b divides c: if not, there are no solutions; if it does, there are …

#### How do you solve Diophantine equations using Euclidean algorithms?

Find a solution to the Diophantine equation 172x + 20y = 1000. Use the Division Algorithm to find d = gcd(172, 20). Use the Euclidean Algorithm to find x* and y* such that d = ax* + by*. Solve for the remainder.

**What is linear congruences explain?**

Generally, a linear congruence is a problem of finding an integer x that satisfies the equation ax = b (mod m). Thus, a linear congruence is a congruence in the form of ax = b (mod m), where x is an unknown integer. In a linear congruence where x0 is the solution, all the integers x1 are x1 = x0 (mod m).

**Which of the following Diophantine equation has no solution?**

Diophantine equations fall into three classes: those with no solutions, those with only finitely many solutions, and those with infinitely many solutions. For example, the equation 6x − 9y = 29 has no solutions, but the equation 6x − 9y = 30, which upon division by 3 reduces to 2x − 3y = 10, has infinitely many.

## What is non linear Diophantine equation?

A non- linear Diophantine equation is every Diophantine equation which is not linear. For instance, the equation $x^2 + 3y^3 = 35$ is a non-linear Diophantine equation.

**Which of the following Diophantine equation is solvable?**

For instance, we know that linear Diophantine equations are solvable.

**Which of the following Diophantine equation is not solvable?**

gcd(6, 51) = 3Hence the equation is not solvable. gcd(33, 14) = 1.

### What are the integer solutions to the linear Diophantine equation?

Then the linear Diophantine equation has a solution if and only if c c. Thus, the values 13 13. Now, for any such ). Then, the above implies the integer solutions to the equation are m m integer. 4x + 7y = 97 4x+7y = 97.

**How do you convert a modular equation to a Diophantine equation?**

Introduce a second variable to convert the modular equation to an equivalent diophantine equarion. So 28x = 38 + 42y for some integers x and y. Simplify to 14 (2x – 3y) = 38. But 2x – 3y is an integer. The left side is always a multiple of 14, but 38 is not. So that equation has no solutions mod 42.

**What is a linear Diophantine tree?**

Suffix Tree. Ukkonen’s Algorithm A Linear Diophantine Equation (in two variables) is an equation of the general form: where a, b, c are given integers, and x, y are unknown integers. In this article, we consider several classical problems on these equations:

#### How do you find the integral solution to a linear equation?

You must first find the greatest common factor of the coefficients in the problem, and then use that result to find a solution. If you can find one integral solution to a linear equation, you can apply a simple pattern to find infinitely many more.