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a) Use the method of direct proof to prove the following statements. i.Prove that the product of two integers, one of the forms 5k₁ + 1 and the other of the form 5k₂ + 1, where k₁ and k₂ are integers, is of the form 5k3 + 1 for some integer k3. \text { ii. Suppose } p, q \in \mathbb{Z} \text {. If } p \mid q \text { then } p^{2} \mid q^{2} \text {. } b) Use the method of proof by contradiction to prove the following statements. \text { i. For all real numbers } x \text { and } y \text {, if } x+y \geq 2 \text {, then either } x \geq 1 \text { or } y \geq 1 \text {. } \text { ii. Prove that } \sqrt{2} \text { is irrational. } Find counterexample to show that the statements are false. i. There exist three integers x, y, z, all greater than 1 and no two equal for which \text { ii. } \quad \forall \text { integer } m \text { and } n, m \times n \geq m+n \text {. } \text { iii. If } r, s \in \mathbb{R} \text {, then }|r+s|=|r|+|s| \text {. } \text { iv. } \forall m \in \mathbb{Z}, \frac{m^{2}}{1+m^{2}} \text { is an integer. }

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