does 54x + 5356y + 645z^2 = 45 have an integer solution?

To determine if the equation \( 54x + 5356y + 645z^2 = 45 \) has integer solutions, we need to analyze the equation for possible integer values of \( x \), \( y \), and \( z \).

Given:
\[ 54x + 5356y + 645z^2 = 45 \]

We can rewrite the equation modulo 3, 5, and 7 to investigate potential integer solutions.

### Modulo 3:
\[ 54x + 5356y + 645z^2 \equiv 0x + 1y + 0z^2 \equiv y \pmod{3} \]
\[ 45 \equiv 0 \pmod{3} \]

So,
\[ y \equiv 0 \pmod{3} \]

### Modulo 5:
\[ 54x + 5356y + 645z^2 \equiv 4x + y + 0z^2 \equiv 4x + y \pmod{5} \]
\[ 45 \equiv 0 \pmod{5} \]

So,
\[ 4x + y \equiv 0 \pmod{5} \]

### Modulo 7:
\[ 54x + 5356y + 645z^2 \equiv 5x + 4y + 1z^2 \pmod{7} \]
\[ 45 \equiv 3 \pmod{7} \]

So,
\[ 5x + 4y + z^2 \equiv 3 \pmod{7} \]

From the modulo analysis, we can use these congruences to check for integer solutions:

1. \( y \equiv 0 \pmod{3} \)
2. \( 4x + y \equiv 0 \pmod{5} \)
3. \( 5x + 4y + z^2 \equiv 3 \pmod{7} \)

### Further Analysis:
Using the modulo conditions, we can test specific values to find integer solutions.

#### Let's try \( z = 2 \):
\[ 5x + 4y + 2^2 \equiv 3 \pmod{7} \]
\[ 5x + 4y + 4 \equiv 3 \pmod{7} \]
\[ 5x + 4y \equiv -1 \pmod{7} \]
\[ 5x + 4y \equiv 6 \pmod{7} \]

Now, substitute \( y \equiv 0 \pmod{3} \):
\[ y = 3k \text{ for integer } k \]

Substitute \( y = 3k \) into the second congruence:
\[ 4x + 3k \equiv 0 \pmod{5} \]
\[ 4x \equiv -3k \pmod{5} \]

Let's find values of \( k \) and \( x \) that satisfy all conditions.

### Simplifying:

To simplify, we'll use the original equation to check specific values:

Trying \( z = 1 \), \( y = 0 \):

\[ 54x + 645 = 45 \]
\[ 54x = -600 \]
\[ x = -\frac{600}{54} \approx -11.11 \]

This is not an integer. Hence, \( z = 1 \) does not yield an integer solution.

Trying \( z = 0 \), \( y = 0 \):

\[ 54x = 45 \]
\[ x = \frac{45}{54} \approx 0.83 \]

This is not an integer either. Similarly, we can see higher values of \( z \) will make \( x \) more non-integral.

After testing various values and seeing no integer solutions, we conclude that the equation \( 54x + 5356y + 645z^2 = 45 \) likely does not have integer solutions based on modulo analysis and specific value testing.

Would you like more details or have any questions?

### Related Questions:
1. How do you solve linear Diophantine equations?
2. What are the properties of modulo arithmetic?
3. Can quadratic terms in equations affect integer solutions?
4. How do we verify if an equation has a solution in number theory?
5. What are some methods to find integer solutions in polynomial equations?
6. How does modulo operation help in simplifying large equations?
7. What are the common techniques to solve equations involving three variables?
8. How do we use the Chinese Remainder Theorem in solving such problems?

### Tip:
When solving equations for integer solutions, modular arithmetic can significantly simplify the process by reducing the problem to simpler congruences.

Explore the analysis of the equation 54x + 5356y + 645z^2 = 45 for integer solutions. This problem involves modular arithmetic and number theory concepts, such as Diophantine equations and the Chinese Remainder Theorem. Suitable for advanced high school students and those interested in number theory.

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