Beginning Activity Beginning Activity 1: Integer Solutions for Linear Equations in One Variable
1.
Does the linear equation \(6x = 42\) have a solution that is an integer? Explain.
2.
Does the linear equation \(7x = -21\) have a solution that is an integer? Explain.
3.
Does the linear equation \(4x = 9\) have a solution that is an integer? Explain.
4.
Does the linear equation \(-3x = 20\) have a solution that is an integer? Explain.
5.
Prove the following theorem:
Theorem 8.20.
Let \(a, b \in \mathbb{Z}\) with \(a \ne 0\text{.}\)
If \(a\) divides \(b\text{,}\) then the equation \(ax = b\) has exactly one solution that is an integer.
If \(a\) does not divide \(b\text{,}\) then the equation \(ax = b\) has no solution that is an integer.