File Edit Viexy Wirdaw Help Q3. Prove that for any two rational numbers $a$ and $b, 5 a^{2}-7 b^{2}+3 a b$ is rational. (10 pts) Q4. Prove that the product of a non-zero rational number and an irrational number is irrational. (10 pts.) Q5. Prove that for any three non-zero integers $a, b$, and $c$, if $a \mid b$ and $a \mid(7 b-c)$ then $a \mid c$. (10 pts) Q6. Prove that for any two integers $m$ and $n, m^{2} n+n^{2} m-2$ is even. (10 pts) Q7. Prove that for any integer $p$, if $3 \mid p^{2}$ then $3 \mid p$. 08. Prove that $\sqrt{3}$ is irrational. (10 pts) 09. Prove by induction that for any integer $n>3, 2^{n}0,6 \mid\left(n^{3}- n ight $ (10 pts) CS.VS. 1431 File Edit Viexy Wirdaw Help Q3. Prove that for any two rational numbers $a$ and $b, 5 a^{2}-7 b^{2}+3 a b$ is rational. (10 pts) Q4. Prove that the product of a non-zero rational number and an irrational number is irrational. (10 pts.) Q5. Prove that for any three non-zero integers $a, b$, and $c$, if $a \mid b$ and $a \mid(7 b-c)$ then $a \mid c$. (10 pts) Q6. Prove that for any two integers $m$ and $n, m^{2} n+n^{2} m-2$ is even. (10 pts) Q7. Prove that for any integer $p$, if $3 \mid p^{2}$ then $3 \mid p$. 08. Prove that $\sqrt{3}$ is irrational. (10 pts) 09. Prove by induction that for any integer $n>3, 2^{n}0,6 \mid\left(n^{3}- n ight $ (10 pts) CS.VS. 1431<><>