Beginning Activity Beginning Activity 1: Exploring Examples where \(\boldsymbol{a}\) Divides \(\boldsymbol{b \cdot c}\)
1.
Find at least three different examples of nonzero integers \(a\text{,}\) \(b\text{,}\) and \(c\) such that \(a \mid \left( {bc} \right)\) but \(a\) does not divide \(b\) and \(a\) does not divide \(c\text{.}\) In each case, compute \(\gcd( {a, b} )\) and \(\gcd( {a, c} )\text{.}\)
2.
Find at least three different examples of nonzero integers \(a\text{,}\) \(b\text{,}\) and \(c\) such that \(\gcd( {a, b} ) = 1\) and \(a \mid ( {bc} )\text{.}\) In each example, is there any relation between the integers \(a\) and \(c\text{?}\)
3.
Formulate a conjecture based on your work in Exercise 1 and Exercise 2.