4.6: Section 6-Exercises

Use the method of direct proof to prove the following statements. 

Exercise $\PageIndex{1}$

If x is an even integer, then $x^2$ is even.

Exercise $\PageIndex{2}$

If x is an odd integer, then $x^3$ is odd.

Exercise $\PageIndex{3}$

If a is an odd integer, then $a^2+3a+5$ is odd.

Exercise $\PageIndex{4}$

Suppose $x,y \in \mathbb{Z}$. If x and y are odd, then xy is odd.

Exercise $\PageIndex{5}$

Suppose $x, y \in \mathbb{Z}$. If x is even, then xy is even.

Exercise $\PageIndex{6}$

Suppose $a, b, c \in \mathbb{Z}$. If $a|b$ and $a|c$, then $a|(b+c)$. 

Exercise $\PageIndex{7}$

Suppose $a, b \in \mathbb{Z}$. If $a|b$, then $a^{2}|b^{2}$.

Exercise $\PageIndex{8}$

Suppose a is an integer. If $5|2a$, then $5|a$.

Exercise $\PageIndex{9}$

Suppose a is an integer. If $7|4a$, then $7|a$.

Exercise $\PageIndex{10}$

Suppose a and b are integers. If $a|b$, then $a|(3b^3-b2+5b)$.

Exercise $\PageIndex{11}$

Suppose $a, b, c, d \in \mathbb{Z}$. If $a|b$ and $c|d$, then $ac|bd$. 

Exercise $\PageIndex{12}$

If $x \in \mathbb{R}$ and $0<x<4$, then $\frac{4}{x(4−x)} \ge 1$. 

Exercise $\PageIndex{13}$

Suppose $x, y \in \mathbb{R}$. If $x^2+5y = y^2+5x$, then $x=y$ or $x+y = 5$.

Exercise $\PageIndex{14}$

If $n \in \mathbb{Z}$, then $5n^2+3n+7$ is odd. (Trycases.)

Exercise $\PageIndex{15}$

If $n \in \mathbb{Z}$, then $n^2+3n+4$ is even. (Trycases.)

Exercise $\PageIndex{16}$

If two integers have the same parity, then their sum is even. (Try cases.)

Exercise $\PageIndex{17}$

If two integers have opposite parity, then their product is even.

Exercise $\PageIndex{18}$

Suppose x and y are positive real numbers. If $x < y$, then $x^{2} < y^{2}$.

Exercise $\PageIndex{19}$

Suppose a, b and c are integers. If $a^{2}|b$ and $b^{3}|c$, then $a^6|c$.

Exercise $\PageIndex{20}$

If a is an integer and $a^{2}|a$, then $a \in \{−1, 0, 1\}$.

Exercise $\PageIndex{21}$

If p is prime and k is an integer for which $0<k<p$, then p divides ${p \choose k}$.

Exercise $\PageIndex{22}$

If $n \in \mathbb{N}$, then $n^{2} = 2{n \choose 2} + {n \choose 1}$. (You may need a separate case for $n = 1$.)

Exercise $\PageIndex{23}$

If $n \in \mathbb{N}$, then ${2n \choose n}$ is even.

Exercise $\PageIndex{24}$

If $n \in \mathbb{N}$ and $n \ge 2$, then the numbers $n!+2, n!+3, n!+4, n!+5, \cdots , n!+n$ are all composite. (Thus for any $n \ge 2$, one can find $n - 1$ consecutive composite numbers. This means there are arbitrarily large “gaps” between prime numbers.)

Exercise $\PageIndex{25}$

If $a, b, c \in \mathbb{N}$ and $c \le b \le a$, then ${a \choose b}{a \choose c} = {a \choose b-c}{a-b+c \choose c}$.

Exercise $\PageIndex{26}$

Every odd integer is a difference of two squares. (Example $7 = 4^{2}-3^{2}$, etc.)

Exercise $\PageIndex{27}$

Suppose $a, b \in \mathbb{N}$. If $gcd(a, b) > 1$, then $b|a$ or b is not prime.

Exercise $\PageIndex{28}$

Let $a, b, c \in \mathbb{Z}$. Suppose a and b are not both zero, and $c \ne 0$. Prove that $c \cdot gcd(a,b) \le  gcd(ca, cb)$.