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Magazine Problem 4
Prove the following statements with contrapositive proof. (In each case, think about how a direct proof would work. In most cases contrapositive is easier.) Suppose \(a, b, c \in \mathbb{Z} .\) If \(a\) does not divide \(b c,\) then \(a\) does not divide \(b\).
Problem 7
Prove the following statements with contrapositive proof. (In each case, think about how a direct proof would work. In most cases contrapositive is easier.) Suppose \(a, b \in \mathbb{Z}\). If both \(a b\) and \(a+b\) are even, then both \(a\) and \(b\) are even.
Problem 12
Prove the following statements with contrapositive proof. (In each case, think about how a direct proof would work. In most cases contrapositive is easier.) Suppose \(a \in Z\). If \(a^{2}\) is not divisible by \(4,\) then \(a\) is odd.
Problem 15
Prove the following statements using either direct or contrapositive proof. Suppose \(x \in \mathbb{Z} .\) If \(x^{3}-1\) is even, then \(x\) is odd.
