24178: HW 2

Problem 6. Is the statement $\quad\exists !\,x\in\mathbb{R} : (x-2=\sqrt{x+7}) \quad$ true or false? Prove your conjecture.

Problem 7. Let $x$, $y$, and $z$ be natural numbers. Prove or disprove: If $x+y$ is even and $y+z$ is even, then $x+z$ is even.

Problem 8. Let $x$, $y$, and $z$ be natural numbers. Prove or disprove: If $x+y$ is odd and $y+z$ is odd, then $x+z$ is odd.

Problem 9. Let $x$ be a natural number. Prove or disprove: If $x^2$ is divisible by 27, then $x$ is divisible by 9.

Problem 10. Let $n$ be a natural number. Show that $\sqrt{n} is a natural number if and only if $n=k^2$ for some natural number $k$.