You need to get exactly -1 from the number 180 using only the numbers -30, -42, and +75. What are the possible solutions?

We are tasked with obtaining exactly \(-1\) from the number 180 by using only the numbers \(-30\), \(-42\), and \(+75\). This means we need to solve the equation:

\[
180 + a \times (-30) + b \times (-42) + c \times (+75) = -1
\]

Where \(a\), \(b\), and \(c\) are integers representing how many times we add or subtract \(-30\), \(-42\), and \(75\), respectively.

The equation becomes:

\[
180 - 30a - 42b + 75c = -1
\]

Rearranging it:

\[
30a + 42b - 75c = 181
\]

We now need to find integer values for \(a\), \(b\), and \(c\) such that this equation holds.

### Step-by-Step Approach:
1. Find values of \(a\), \(b\), and \(c\) that satisfy the equation \(30a + 42b - 75c = 181\).
2. Since these numbers have common factors (e.g., 30 and 42 share a factor of 6), we'll experiment with integer multiples to solve for the variables.

Let's attempt solving it step by step.

The equation \(30a + 42b - 75c = 181\) simplifies to:

\[
a = -\frac{7b}{5} + \frac{5c}{2} + \frac{181}{30}
\]

This suggests that we need to find specific integer values of \(b\) and \(c\) such that \(a\) is also an integer.

Let's now test integer values of \(b\) and \(c\) to find possible solutions that make \(a\) an integer.It appears that within the range of values tested for \(b\) and \(c\), no integer solutions for \(a\) were found. This suggests that there may be constraints or other methods needed to find valid solutions.

Would you like to explore a different approach or expand the range of values for testing? Additionally, let me know if you'd like more details on the process or any further questions. 

Here are some related questions:
1. How do we solve Diophantine equations systematically?
2. What is the role of modular arithmetic in finding integer solutions?
3. Can we generalize this method for equations with multiple variables?
4. How does linear combination work in solving equations like this?
5. What other tools can be used to solve integer-based equations?

**Tip:** When working with integer solutions, focus on common divisors and simplify terms as much as possible.

Explore how to solve the linear Diophantine equation to get -1 from 180 using -30, -42, and +75. This problem involves algebraic concepts and linear combinations, perfect for students in Grades 10-12. Learn the step-by-step process of solving this equation.

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