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\begin{center} \textbf{Theorems about the integers.} \end{center}

For these theorems you must use only the Axioms of the integers and the rules of logic and sets; the axioms of the integers are stated in the file AxiomsOfTheIntegers.

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Definition: If $a \in \mathbb{R}$ then: $a^0 = 1$; $a^1 = a$; $a^{n+1} = a^n \cdot a$.

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Definition: $2 = 1 + 1$.  [Note that this is the definition of the symbol ``$2$''; the symbols $3, 4, ... $ are defined inductively similarly.]

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Theorem 3.1.  Suppose that each of $a$, $b$ and $c$ is an integer.

\qquad a. $(a+b) \cdot c = a \cdot c + b \cdot c$.

\qquad b. $a +(b+c) = (c+a) + b$.

\qquad c. $a \cdot(b \cdot c) = (c \cdot a) \cdot b$.

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Exercise 3.1x. Suppose that each of $a$, $b$ and $c$ is an integer.

\qquad (i.) List all the variations on Theorem 3.1a and prove a couple of them.

\qquad (ii.) List all the variations on Theorem 3.1b and prove a couple of them; define $a+b+c$.

\qquad (iii.) List all the variations on Theorem 3.1b and prove a couple of them; define $a \cdot b \cdot c$.

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Notational convention: By theorems similar to the above (see exercise 3.1x.) and the associativity and commutativity axioms, $a + b + c$ can now be defined.  The quantity $a \cdot b \cdot c$ can be similarly defined.

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Notational convention: $ab$ means $a \cdot b$.

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Theorem 3.1 (continued):

\qquad d. The additive and multiplicative identities are both
unique.  (i.e. no number other than $0$ is the additive identity and
similarly for the multiplicative identity.)

\qquad e. $(a+b)^2 = a^2 + 2ab + b^2$.

\qquad f. $0 \cdot a = 0$.

\qquad g. If $a+b = 0$ and  $a+c = 0$ then $b=c$.

\qquad h. $(-1)\cdot a = -a$. [Hint: use f and g.]

\qquad i. $-(-a)=a$.

\qquad j. $ (-a)\cdot b = -(ab) = a \cdot (-b)$.

\qquad k. $(-a)\cdot (-b) = a \cdot b$.

\qquad l. $-0 = 0$.

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Notational convention:  $a>b$ means $b<a$.

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Theorem 3.2.  Suppose that each of $a$, $b$ and $c$ is an integer.

\qquad a. If $a < b$ and $0>c$ then $ac > bc$.

\qquad b. If $a \ne 0$, then $a^2 > 0$.

\qquad c.  If $ab =0$ then either $a=0$ or $b=0$.

\qquad d.  If $a > 0$ then $-a < 0$.

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Theorem 3.3. There is no integer between $0$ and $1$.

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Definition.  If $S$ is a subset of $\mathbb{Z}$ then $\ell$ is the least element of $S$ means $\ell \in S$ and if $x \in S$ then $\ell \le x.$

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Theorem 3.4.  If $S \subset \mathbb{N}$ and $S$ is non-empty then $S$ has a least element.

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\textbf{Note: From this point on you may assume all the algebraic manipulations about the integers, that follow from the axioms about the integers $\mathbb{Z}$, that you have learned in your previous mathematics classes.}

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\begin{center} \textbf{Divisibility.} \end{center}

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\textbf{Note: In this section you may not use fractions since they (and for that matter real numbers) have not yet been defined.}

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Definition.  If $a$ and $b$ are integers then $a$ is said to divide
$b$ if and only if there is an integer $q$ so that $b$=$aq$.  The
standard notation is: $a|b$; this is read as, ``$a$ divides $b$.''

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Theorem 3.5.  If each of $a$, $b$, and $c$ is an integer so that
$a|b$ and $b|c$, then $a|c$.

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Theorem 3.6.  If each of $a$, $b$ and $c$ is an integer so that
$a|b$ and $a|c$ then for arbitrary integers $x$ and $y$,
$a|(xb+yc)$.

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Theorem 3.7.  [The division algorithm.] If each of $a$ and $b$ is an
integer with $0<b$ then there exist unique integers $q$ and $r$ with
$0 \le r < b$ so that:

$$a=bq+r.$$

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In the following exercises and theorems assume, as usual, that the
quantities are appropriately defined.

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Exercise 3.1.  Find integers $a$, $b$, and $c$ so that $a|bc$ but
$a\nmid b$ and $a\nmid c$.

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Theorem 3.8.  If $c \ne 0$ then  $a|b$ iff $ac|bc$.

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Theorem 3.9.  If $a>0$, $b>0$ and $a|b$ then $a\le b$.











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