Section1.7Examples involving divisibility
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Theorem1.7.1Division Algorithm
For any integers \(a,b\) with \(a \not= 0\text{,}\) there exists unique integers \(q\) and \(r\) for which
\begin{equation*}
b = aq + r, \quad 0 \le r \lt \abs{a}.
\end{equation*}
The intger \(b\) is called the dividend, \(a\) the divisor, \(q\) the quotient, and \(r\) the remainder.
Saved for a later chapter.
Definition1.7.2Greatest Common Divisor
Let \(a,b,c \in \mathbb{Z}\) be nonzero. We say \(c\) is a common divisor of \(a\) and \(b\) if and only if \(c\) divides \(a\) and \(c\) divides \(b\text{.}\) The greatest common divisor of \(a,b\) is denoted \(\gcd (a,b)\text{,}\) and is the common divisor of \(a,b\) which is greater than every other common divisor.
Theorem1.7.3
For any \(a,b,x,y \in \mathbb{Z}\text{,}\) any divisor of \(a,b\) also divides \(ax+by\text{.}\) \(ax+by\) is called a linear combination of \(a,b\text{.}\)
Let \(a,b,x,y \in \mathbb{Z}\) be arbitrary. Let \(c\) be any divisor of \(a\) and \(b\) so there exist integers \(r,s\) so that
\begin{equation*}
a = rc, \quad b = sc.
\end{equation*}
Thus
\begin{equation*}
ax + by = (rc)x + (sc)y = c (rx+sy).
\end{equation*}
Since \(rx+sy \in \mathbb{Z}\text{,}\) we have shown \(c\) divides \(ax+by\text{.}\)
The following lemma is the first proof we encounter where the key idea to get started is not obvious, even after a bit of playing around. At this point in the course, you would not be expected to come up with a proof like this, although later in the course you would.
Lemma1.7.4
For any nonzero \(a,b \in \mathbb{Z}\text{,}\) the smallest positive linear combination of \(a,b\) is a common divisor.
Let \(a,b\) be arbitrary nonzero integers. Assume \(x,y \in \mathbb{Z}\) so that \(s := ax+by\) is positive, but as small as possible (i.e., it is the smallest positive linear combination of \(a,b\)).
By Division Algorithm, we may write \(a = sq + r\) for some \(q,r \in \mathbb{Z}\) with \(0 \le r \lt \abs{s} = s\text{.}\) Now
\begin{equation*}
r = a - sq = a - (ax + by)q = a(1-xq) + b(yq)
\end{equation*}
is another (nonnegative) linear combination of \(a,b\text{.}\) But we know \(r\) cannot be positive, otherwise it would violate the minimality of \(s\text{.}\) Therefore, \(r = 0\) and hence \(a = sq\text{.}\) Thus \(s\) divides \(a\text{.}\) A nearly identical argument shows \(s\) divides \(b\text{,}\) and so \(s\) is a common divisor.
Theorem1.7.5
For any nonzero integers \(a,b\text{,}\) their greatest common divisor is their smallest linear combination.
Let \(a,b\) be arbitrary nonzero integers and let \(s\) denote their smallest positive linear combination. By Lemma 1.7.4 \(s\) is a common divisor of \(a,b\text{.}\) Let \(x\) be any other positive common divisor of \(a,b\text{.}\) By Theorem 1.7.3, we know \(t\) divides \(s\text{,}\) and so there is some (positive) integer \(k\) for which \(s = tk\text{.}\) Since \(k \in \mathbb{N}\text{,}\) we have \(k \ge 1\) and so \(s = tk \ge t\text{.}\) Therefore \(s\) is greater than any other common divisor and hence \(s = \gcd(a,b)\text{.}\)
Definition1.7.6Relatively Prime, Coprime
We say nonzero integers \(a,b\) are relatively prime, or coprime, if \(\gcd(a,b) = 1\text{.}\)
Theorem1.7.7
For any relatively prime \(a,b \in \mathbb{Z}\) (necessarily nonzero) and for any \(c \in \mathbb{Z}\) there exist \(x,y \in \mathbb{Z}\) so that \(ax+by = c\text{.}\)
Suppose \(a,b \in \mathbb{Z}\) are relatively prime. By Theorem 1.7.5 there exist \(r,s\) so that \(ar+bs = 1\text{.}\) Then
\begin{equation*}
a(rc)+b(sc) = (ar+bs)c = c.
\end{equation*}
Setting \(x = rc, y = sc \in \mathbb{Z}\) proves the result.
Lemma1.7.8Euclid's Lemma
Let \(a,b,p \in \mathbb{Z}\) with \(p\) prime. If \(p\) divides \(ab\text{,}\) then either \(p\) divides \(a\) or \(p\) divides \(b\text{.}\)
Let \(a,b,p \in \mathbb{Z}\) with \(p\) prime and suppose \(p\) divides \(ab\text{.}\) Thus \(ab = pk\) for some integer \(k\text{.}\) Note that \(\gcd(a,p)\) can be only either \(1\) or \(p\) since \(p\) has no other divisors.
Case 1: \(\gcd(a,p) = p\).
Then \(p\) divides \(a\) and we are finished.
Case 2: \(\gcd(a,p) = 1\).
By Theorem 1.7.5 there exist \(x,y \in \mathbb{Z}\) so that \(ax + py = 1\text{.}\) Multiplying both sides by \(b\text{,}\) we obtain
\begin{equation*}
b = (ax+py)b = abx+pyb = pkx+pyb = p(kx+yb).
\end{equation*}
Therefore \(p\) divides \(b\text{.}\)