The book I am using lists something like $x \equiv2-(5/4) \equiv 2+5 \equiv 7 \pmod {25}$
I dont understand why $(-5/4)$ seemingly turns into $5$ mod $25$
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2$$\dfrac{5}{4}\equiv\dfrac{5\times 6}{4\times 6}\equiv\dfrac{30}{24}\equiv \dfrac{5}{-1}\equiv -5 $$ – AgentS Apr 09 '15 at 02:57
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Im curious as to where the 6 comes from here, Why can you just multiply everything by 6? – user2327195 Apr 09 '15 at 03:04
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we can just multiply both sides by $6$ because $x\equiv y\pmod{n} \implies cx \equiv cy \pmod{n}$ : $$4x\equiv 5 \pmod{25} \implies 6\times 4x \equiv 6\times 5 \pmod{25}$$ – AgentS Apr 09 '15 at 03:09
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Dividing by 4 is OK mod 25 because 4 and 25 are relatively prime. – GEdgar Apr 09 '15 at 14:59
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Hint: Because $4\cdot 19\equiv 76\equiv 1 \pmod{25}$, we have that the inverse of $4$ is $19 \pmod{25}$. That is $\frac14 \equiv 19 \pmod{25}$.
paw88789
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Yes. $76\equiv 1 \pmod{25}$, because they differ by a multiple of $25$. – paw88789 Apr 09 '15 at 09:37
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${\rm mod}\ 25\!:\ 4x \equiv -5\equiv 20\iff x\equiv 5,\, $ via cancel $4,\,$ valid by $\,(4,25)=1\,\Rightarrow\,4^{-1}\,$ exists.
Equivalently $\ x\equiv \dfrac{-5}4\equiv \dfrac{20}4\equiv 5$
Beware $ $ Modular fraction arithmetic is well-defined only for fractions with denominator coprime to the modulus. See here for further discussion.
Bill Dubuque
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