(Solved): **please help with 3, 6, and 7!** For each of the following statements: (a) Is the statement true ...

**please help with 3, 6, and 7!**

For each of the following statements: (a) Is the statement true or false? (b) If it is false, disprove it. (c) If it is true, say what proof method is most appropriate. Then prove the statement from the definitions (of rational, even, divisibility, mod, etc.) using that method. 1. There exists a prime number $p$ so that $p^2?p+11$ is also prime. 2. For all real numbers $x$, the number $?x$ is negative. 3. For all integers $a,b$, and $c$, if $a,b$, and $c$ are consecutive, then $a+b+c$ is even. 4. For all integers $a,b,c$, and $d$, if $a?c$ and $b?d$, then $ab?cd$. 5. For any integer $n,n^2+5$ is not divisible by 5 . Hint: Every integer is either . . or $\dots$. 6. For all integers $a,b$, and $c$, if $a?bc$, then $a?b$. 7. There is no integer $n$ so that $n?1\mathrm{mod}3$ and $n?2\mathrm{mod}3$. Hint: Use a method from 4.7. ${}^1$