명제의 증명과 절대부등식

Give at least four different examples of integers $a$, $b$, and $c$ with $a \ne 0$ such that $a$ divides $b$ and $a$ divides $c$.

Construct a know-show table for the following proposition: For all integers $a$, $b$, and $c$ with $a \ne 0$, if $a$ divides $b$ and $a$ divides $c$, then $a$ divides $(b + c)$.

Use a counterexample to prove the following statement is false.

For all integers

a

and

b

, if 5 divides

a

or 5 divides

b

, then 5 divides

(5a + b)

.

Prove each of the following statements:

For all integers $a$, $b$, and $c$ with $a \ne 0$, if $a \mid b$ and $a \mid c$, then $a \mid \left(b-c \right)$.

Prove each of the following statements:

For each integer $a$, if 4 divides $(a - 1)$, then 4 divides $\left( a^2 - 1 \right)$.

Hint: If 4 divides $(a - 1)$, then there exists an integer $k$ such that $a - 1 = 4k$ and so $a = 4k + 1$. Use algebra to rewrite $\left( a^2 - 1 \right) = (4k + 1)^2 - 1$.

For each of the following, use a counterexample to prove the statement is false.

For each odd natural number $n$, if $n > 3$, then 3 divides $\left( n^2 - 1 \right)$.

For each of the following, use a counterexample to prove the statement is false.

For each natural number $n$, $\left( 3 \cdot 2^n + 2 \cdot 3^n + 1 \right)$ is a prime number.

For each of the following, use a counterexample to prove the statement is false.

For all real numbers $x$ and $y$, $\sqrt{x^2 + y^2} > 2xy$.

For each of the following, use a counterexample to prove the statement is false.

For each integer $a$, if 4 divides $\left( a^2 - 1 \right)$, then 4 divides $(a - 1)$.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For all integers $a$, $b$, and $c$ with $a \ne 0$, if $a \mid b$, then $a \mid \left(bc \right)$.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For all integers $a$ and $b$ with $a \ne 0$, if $6 \mid \left(ab \right)$, then $6 \mid a$ or $6 \mid b$.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For all integers $a$, $b$, and $c$ with $a \ne 0$, if $a$ divides $(b - 1)$ and $a$ divides $(c - 1)$, then $a$ divides $(bc - 1)$.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For each integer $n$, if 7 divides $\left( n^2 - 4 \right)$, then 7 divides $(n - 2)$.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For every integer $n$, $4n^2 + 7n + 6$ is an odd integer.

Hint: Make sure you first try some examples. How do you prove that an integer is an odd integer?

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For every odd integer $n$, $4n^2 + 7n + 6$ is an odd integer.

Hint: The following algebra may be useful.

$$4 \left( 2m + 1 \right)^2 + 7 \left( 2m + 1 \right) + 6 = 16m^2 + 30m + 17$$

.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For all integers $a$, $b$, and $d$ with $d \ne 0$, if $d$ divides both $a-b$ and $a+b$, then $d$ divides $a$.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For all integers $a$, $b$, and $c$ with $a \ne 0$, if $a \mid (bc)$, then $a \mid b$ or $a \mid c$.

Complete the following.

If $x$ and $y$ are integers and $xy=1$, explain why $x=1$ or $x=-1$.

Prove the following proposition:

Let

a

be an integer. If there exists an integer

n

such that

a \mid \left(4n+3 \right)

and

a \mid \left( 2n+1 \right)

, then

a = 1

or

a = - 1

.

Hint: Use the fact that the only divisors of 1 are 1 and $-1$.

$\left( 4n + 3 \right) - 2 \left( 2n + 1 \right) = 1$.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For each integer $a$, if there exists an integer $n$ such that $a$ divides $(8n + 7)$ and $a$ divides $(4n + 1)$, then $a$ divides 5.