Key Concepts and Formulas

• Diophantine Equation: An equation where only integer solutions are of interest.
• Non-negative Integer Solutions: Solutions to an equation where the variables can only take non-negative integer values (0, 1, 2, ...).
• Iterative Approach: When dealing with multiple variables, iterate through possible values of one variable to simplify the problem.

Step-by-Step Solution

Step 1: Isolate x and establish bounds for z.

We are given the equation $x + 2y + 3z = 42$, where $x, y, z \ge 0$ are integers. We want to find the number of possible triples $(x, y, z)$ that satisfy this equation. First, isolate $x$: $x = 42 - 2y - 3z$ Since $x \ge 0$, we have $42 - 2y - 3z \ge 0$, which implies $2y + 3z \le 42$. Since $y \ge 0$, we must have $3z \le 42$, which means $z \le 14$. Also, since $z \ge 0$, we know $0 \le z \le 14$.

Step 2: Iterate through possible values of z.

We will now iterate through possible integer values of $z$ from 0 to 14. For each value of $z$, we will determine the possible values of $y$. Since $2y + 3z \le 42$, we have $2y \le 42 - 3z$, so $y \le \frac{42 - 3z}{2}$. Since $y \ge 0$, we have $0 \le y \le \frac{42 - 3z}{2}$.

• For a fixed $z$, the number of possible integer values for $y$ is $\left\lfloor \frac{42 - 3z}{2} \right\rfloor + 1$. (We add 1 to include the case where $y=0$).
• Since $x = 42 - 2y - 3z$, for each pair of $y$ and $z$ we will have exactly one value of $x$ satisfying the equation $x+2y+3z=42$. Thus, the number of solutions is the sum of the number of possible values for $y$ for each value of $z$.

Step 3: Calculate the number of solutions for each z and sum them up.

We need to calculate the sum: $\sum_{z=0}^{14} \left( \left\lfloor \frac{42 - 3z}{2} \right\rfloor + 1 \right)$

We can expand this summation as follows: $\left( \left\lfloor \frac{42}{2} \right\rfloor + 1 \right) + \left( \left\lfloor \frac{42-3}{2} \right\rfloor + 1 \right) + \left( \left\lfloor \frac{42-6}{2} \right\rfloor + 1 \right) + \dots + \left( \left\lfloor \frac{42-3(14)}{2} \right\rfloor + 1 \right)$ $= \sum_{z=0}^{14} \left( \left\lfloor \frac{42 - 3z}{2} \right\rfloor + 1 \right) = \sum_{z=0}^{14} \left\lfloor \frac{42 - 3z}{2} \right\rfloor + \sum_{z=0}^{14} 1 = \sum_{z=0}^{14} \left\lfloor \frac{42 - 3z}{2} \right\rfloor + 15$

Now we can expand the first summation term by term: $\left\lfloor \frac{42}{2} \right\rfloor + \left\lfloor \frac{39}{2} \right\rfloor + \left\lfloor \frac{36}{2} \right\rfloor + \left\lfloor \frac{33}{2} \right\rfloor + \left\lfloor \frac{30}{2} \right\rfloor + \left\lfloor \frac{27}{2} \right\rfloor + \left\lfloor \frac{24}{2} \right\rfloor + \left\lfloor \frac{21}{2} \right\rfloor + \left\lfloor \frac{18}{2} \right\rfloor + \left\lfloor \frac{15}{2} \right\rfloor + \left\lfloor \frac{12}{2} \right\rfloor + \left\lfloor \frac{9}{2} \right\rfloor + \left\lfloor \frac{6}{2} \right\rfloor + \left\lfloor \frac{3}{2} \right\rfloor + \left\lfloor \frac{0}{2} \right\rfloor$ $= 21 + 19 + 18 + 16 + 15 + 13 + 12 + 10 + 9 + 7 + 6 + 4 + 3 + 1 + 0 = 154$ Thus, the total number of solutions is $154 + 15 = 169$.

Common Mistakes & Tips

• Forgetting the +1: Remember to add 1 when counting the number of possible values for $y$, to account for the case where $y=0$.
• Integer Division: Be careful with the floor function. $\lfloor x \rfloor$ gives the largest integer less than or equal to $x$.
• Systematic Approach: Iterating through possible values of one variable and solving for the others is a reliable strategy for these types of problems.

Summary

We found the number of non-negative integer solutions to $x + 2y + 3z = 42$ by isolating $x$, establishing bounds for $z$, and then iterating through possible values of $z$. For each value of $z$, we calculated the number of possible values for $y$ and summed these up to find the total number of solutions. This resulted in a total of 169 solutions.

Final Answer

The final answer is \boxed{169}.