I have a question about congruences involving fractions.(adsbygoogle = window.adsbygoogle || []).push({});

For integers a and b the following is defined:

a and b are congruent modulo m (m is a natural number) if there exists

an integer k such that k*m = a-b

[itex]a \equiv b (\mbox{mod } m) \Longleftrightarrow \exists k \in \mathbb{Z} : km = a-b[/itex]

For example:

[itex]13 \equiv 4 (\mbox{mod } 9)[/itex] because [itex]1 \cdot 9 = 13-4[/itex]

On the Wolfram mathworld website there are other examples in (8):

[itex]2 \cdot 4 \equiv 1 (\mbox{mod } 7)[/itex]

[itex]3 \cdot 3 \equiv 2 (\mbox{mod } 7)[/itex]

[itex]6 \cdot 6 \equiv 1 (\mbox{mod } 7)[/itex]

So far, so good.

But then in (9) they write:

[itex]\frac{1}{2} \equiv 4 (\mbox{mod } 7)[/itex]

[itex]\frac{1}{4} \equiv 2 (\mbox{mod } 7)[/itex]

[itex]\frac{2}{3} \equiv 3 (\mbox{mod } 7)[/itex]

[itex]\frac{1}{6} \equiv 6 (\mbox{mod } 7)[/itex]

which I don't understand.

At first I thought that for fractions a and b the definition is just extended:

[itex]a \equiv b (\mbox{mod } m) \Longleftrightarrow \exists k \in \mathbb{Z} : km = a-b[/itex]

with a and b fractions (instead of just integers).

But the definition of congruence for fractions must be different since

there is no [itex]k \in \mathbb{N}[/itex] such that

[itex]\frac{1}{2} - 4 = k \cdot 7:[/itex]

[itex]\frac{1}{2} - 4 = k \cdot 7[/itex]

[itex]\Rightarrow \frac{1}{2} - \frac{8}{2} = k \cdot 7[/itex]

[itex]\Rightarrow -\frac{7}{2} = k \cdot 7[/itex]

[itex]\Rightarrow k=-\frac{1}{2}[/itex]

My questions:

a) How are congruences defined for fractions? And why is (9) correct?

b) Does (8) imply (9) ?

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# Congruences with fractions

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