View source for 23666: 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 $k$.

[[Image:Alice3.gif]]