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\handout{miniquiz1}{Mini-Quiz 1}{Sept 15, 2000}

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\item Write your name:
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%Rosen 2.3 15b,c 
\item 
1. Determine whether the following sets of integers are pairwise relatively
prime.
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a) (14, 15, 21)
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This set of integers is not pairwise relatively prime, since 7 divides both 14 and 21, and 3 divides both 15 and 21.
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b) (12, 17, 31, 37)
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This set of integers is pairwise relatively prime, because for any two integers in the set, their greatest common divisor is 1.
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%Rosen 2.3 37
\item
Show that $ac\equiv bc$ (mod m), where a,b,c, and m are integers
such that $m\geq 2$, does not necessarily imply that $a\equiv b$ (mod $m$).
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Proof by counterexample:\\
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There are many possible counterexamples.  (The important thing is to choose c, m such that gcd(m,c)$>$1.)  Let m=4 and c=2.
Then, if we let a=5 and b=15:\\
ac mod m = (10 mod 4) = 2, and\\
bc mod m = (30 mod 4) = 2.\\
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However, $5\not \equiv 15$ (mod 4).
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\end{enumerate}
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