# Help Needed: Solving Integer Equations

• Mattofix
In summary, the conversation discusses finding integers that satisfy linear Diophantine equations and the use of the Euclidean algorithm and factoring to solve them. It also mentions the need for a particular solution and the general solution for these types of equations. Finally, it is noted that if the constant term is not divisible by the greatest common divisor of the coefficients, there is no solution.
Mattofix

## Homework Statement

I don't know where to start on this question - could someone please point me in the right direction so i can look up the method.

Q: Find integers x, y such that 45x + 63y = 99. Can we find integers s, t such that 45s +65t = 80? Either find them or prove they cannot exist.

What have you tried?

when x = 5 and y = -2 , but that was just randomly trying things. is there any method or rule to this?

Look up the Euclidean algorithm. Also, if there are common factors in all the terms, it might be in your best interest to factor them out.

these equations are just a equation of a linear line! both of them; so you can just plot the line and say you will have infinity answers for x and y which have to be on the line. considering both lines: of course these can encounter with each other and as they are not parallel their is exactly one point that is in their intercept (if you deal with x, y).

Mattofix said:

## Homework Statement

I don't know where to start on this question - could someone please point me in the right direction so i can look up the method.

Q: Find integers x, y such that 45x + 63y = 99. Can we find integers s, t such that 45s +65t = 80? Either find them or prove they cannot exist.

The equations $a\,x+b\,y=c$ are called Linear Diophantine Equations. If you acn find a particular solution $(x,y)=(x_o,y_o)$ then you can find the general solution by writting $x=x_o+\lambda\,t,\,y=y_o+\mu\,t$. Plugging these to the original equation you determinate the values of $\lambda,\,\mu$.

You can easily proof that if $c$ is not divisible by $gcd(a,b)$ then there is no solution.

## What are integer equations?

Integer equations are mathematical expressions that contain only whole numbers (positive, negative, or zero) and operations such as addition, subtraction, multiplication, division, and exponentiation. They can also include variables, which represent unknown values.

## What are some common types of integer equations?

Some common types of integer equations include linear equations, quadratic equations, exponential equations, and logarithmic equations. Each type has its own specific form and method for solving.

## How do I solve integer equations?

To solve an integer equation, you need to isolate the variable on one side of the equation and simplify the other side using the order of operations. This can be done by performing inverse operations (e.g. adding and subtracting) or by using algebraic rules (e.g. multiplying both sides by the reciprocal).

## What are some strategies for solving difficult integer equations?

Some strategies for solving difficult integer equations include factoring, completing the square, and using the quadratic formula. It is also helpful to check your solution by plugging it back into the original equation to make sure it satisfies the equation.

## Are there any online resources or tools for solving integer equations?

Yes, there are many online resources and tools available for solving integer equations. Some popular websites include WolframAlpha, Symbolab, and Mathway. These sites offer step-by-step solutions and can also solve more complex equations that may be difficult to solve by hand.

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