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

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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.