Chapter 1.7, Problem 3E

Prove that
(a) $\text{5}{n}^{\text{2}}+\text{3}n+\text{4}$ is even, for all integers n.
(b) for all integers n, if $\text{5}n+\text{1}$ is even, then $\text{2}{n}^{\text{2}}+\text{3}n+\text{4}$ is odd.
(c) the sum of five consecutive integers is always divisible by 5.
(d) $n^{\text{3}}-n$ is divisible by 6, for all integers n.
(e) $(n^{\text{3}}-n)(n+\text{2})$ is divisible by 12, for all integers n.
(f )every four-digit palindrome number is divisible by 11. (A palindrome number is a number that reads the same forward and backward.)
(g) ifa, b, and c are real numbers and $(a-bi)(c-ci)=\text{1}-i$ , then $ac=\text{1}$ .
(h) ifp is a prime integer, then $p+\text{19}$ is composite.
(i) ifn is a natural number and r is the remainder when n is divided by 3, then if $t=\text{5}$ or 11, the sum $r^{\text{2}}+r+t$ is prime.
(j) ifm and n are natural numbers, $n<\text{4}$ , and $(\text{2}-n)(\text{2}-m)>\text{2}(m-n)$ , then $m=\text{1}$ or $n=\text{3}$ .
(k) if $n>\text{2}$ is an even natural number, then ${\text{2}}^n-\text{1}$ is not prime.
(l) ifS is a set of real number such that a, b are in S, and if $x\le a$ and $x\le b$ for every element x of S, then $a=b$ .
(m) ifa and b be integers and b is odd, then 1 and -1 are not solutions of the equation $a{x}^{\text{4}}+b{x}^{\text{2}}+a=0$ .
(n) if two nonvertical lines have slopes whose product is -1, then the lines are perpendicular.