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\begin{center} \textbf{Prime Numbers and the Fundamental Theorem of Arithmetic.}

\textbf{Hints to selected theorems.}
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Theorem* 5.1.  Suppose that each of $a$ and $b$ is a positive number.  If $a|b$ and $b|a$ then $a=b$.

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Hint: Use thm 3.9. 

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Theorem* 5.2.  If $n>1$ is a positive integer then there exists a prime
number $p$ so that $p|n$.

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Hint.  Use The 3.4 and consider two cases: (1) $n$ is prime; (2) $n$ is composite.  

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Theorem* 5.3.  The set of prime numbers in infinite.

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Hint. Argue that if $p$ and$q$ are primes, then neither divides $pq+1$.


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Theorem* 5.8. Suppose that each of $a$ and $b$ is a positive
integer and $d=gcd(a,b)$.  Then there exists integers $x$ and $y$ so that:

$$d = ax + by.$$

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Hint: Let $S =\{ax +by | x, y \in \mathbb{Z} \mbox{ and } (ax + by) > 0 \}$ and use thm 3.4.  (Recall the proof of the division algorithm theorem.)

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Theorem* 5.9.  Suppose that each of $a$ and $b$ is an integer and at
least one of them is not $0$.  Let $S = \{na+mb| n\in \mathbf{Z},
m\in \mathbf{Z}, 0<na+mb \}$.  Then $gcd(a,b)$ is the least element
of the set $S$.

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Hint: Let $S =\{ax +by | x, y \in \mathbb{Z} \mbox{ and } (ax + by) > 0 \}$ and use thm 3.4 and thm 5.8.

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Theorem* 5.10. Let $a$ and $b$ be integers at least one of which is not $0$.  Then $a$ and $b$ are relatively prime if and only if there exist integers $x$ and $y$ so that:
$$ax + by = 1.$$

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Hint: Use Thm 5.8.

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Theorem 5.18'.  Suppose that $p$ is a prime and $n$ is a positive integer greater than one.  Then there is a unique non-negative integer $k$ so that $n = p^k q$ for some integer $q$ and $p \nmid q$ (i.e. $p$ does not divide $q$).  (Note that $n$ can be $0$ in many cases.) [Note that, if not proven separately, this is also a corollary of the Fundamental Theorem of Arithmetic.]

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Hint: Use Thm 5.2.


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