24178: HW 2
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'''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 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 $k$. | + | '''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]] | [[Image:Alice3.gif]] | ||
Latest revision as of 13:40, 2 February 2017
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 $k$.
