If x and y are integers, then the equation 5x + 18y = 63 has

We are tasked with analyzing the equation \( 5x + 18y = 63 \), where \( x \) and \( y \) are integers. Specifically, we want to know if there are any integer solutions, and if so, how many.

This type of equation is called a **Diophantine equation** (linear Diophantine equation, to be precise). The general form is:

\[
ax + by = c
\]

For such an equation to have integer solutions, the greatest common divisor (gcd) of \( a \) and \( b \) must divide \( c \). In this case, the equation is:

\[
5x + 18y = 63
\]

### Step 1: Find gcd of 5 and 18
To determine if this equation has integer solutions, we need to find the gcd of 5 and 18. Since 5 is a prime number and does not divide 18, their gcd is:

\[
\gcd(5, 18) = 1
\]

### Step 2: Check if gcd divides 63
Since \( \gcd(5, 18) = 1 \) and 1 divides any integer, it follows that the equation **has integer solutions**.

### Step 3: Parametric Solution
The general solution to the Diophantine equation can be expressed in terms of a parameter \( t \) once we find one particular solution. We can solve for \( y \) and express \( x \) in terms of \( y \), or vice versa, using the method of the extended Euclidean algorithm.

Would you like me to proceed with finding the specific integer solutions or details on parametric solutions?

### Related Questions:
1. How do we use the extended Euclidean algorithm to find integer solutions?
2. What are Diophantine equations and their general characteristics?
3. What is the role of the gcd in determining the existence of solutions for linear equations?
4. How do we derive parametric solutions for two-variable linear equations?
5. What are the practical applications of solving Diophantine equations?

**Tip:** In Diophantine equations, finding one solution allows us to express all possible solutions parametrically.

Explore how to solve the Diophantine equation 5x + 18y = 63 using number theory concepts like the greatest common divisor (GCD) and the extended Euclidean algorithm. Learn how to find integer solutions step by step, suitable for students in Grades 10-12.

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