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Suppose I have this:

$\frac{6^{666}}{2^{6}}$ (mod $125$)

I saw it is possible to reduce only the numerator's power modulo Euler's phi function. Can someone explain why is that possible?

It is essentially this:

$\frac{6^{666 \space (mod \phi(125))}}{2^{6}}$ (mod $125$)

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  • $\begingroup$ What's the question? There's no need to reduce the exponent in $2^6$ as it is already small. $\endgroup$ – lulu Jan 28 at 17:06
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    $\begingroup$ Can you clarify your question? It really isn't clear what you are asking. $\endgroup$ – lulu Jan 28 at 17:23
  • $\begingroup$ My question is why it works? $\endgroup$ – Michael Munta Jan 28 at 17:25
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    $\begingroup$ Why what works? $\endgroup$ – lulu Jan 28 at 17:27
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    $\begingroup$ @MichaelMunta If, say, you had $\frac {6^{666}}{2^{501}}$ you could use Euler to write that as $\frac {6^{66}}{2^1}\pmod {125}$. $\endgroup$ – lulu Jan 28 at 17:41
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It's valid to mod out the arguments of a fraction - just like it is for the arguments sums and products.

If $(B,n)= 1\,$ then $\bmod n\!:\,\ \begin{align}A\equiv a\\ B\equiv b\end{align}\,\Rightarrow\, \dfrac{A}B\,\overset{\rm def}\equiv A\cdot B^{-1}\equiv a\cdot b^{-1}\,\overset{\rm def}\equiv\, \dfrac{a}b$

So the answer to your question as to "why it works" is that unwinding the definition of a fraction yields a composition of a product and inverse operation - and those operations are compatible with modular aithmetic hence so too is their composition (modular "division" by units = invertibles = integers $B$ coprime to the modulus).

See this answer for much further discussion.

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  • $\begingroup$ So if $B \equiv b$ then also $B^{-1} \equiv b^{-1}$? $\endgroup$ – Michael Munta Jan 28 at 19:29
  • $\begingroup$ @Michael Yes, multiply $\,B\equiv b\ $ by $\ b^{-1}B^{-1}\ $ [assuming $\,(B,n)=1,\,$ so also $\,(b,n)=(B,n)=1$] $\ \ $ $\endgroup$ – Bill Dubuque Jan 28 at 20:06
  • $\begingroup$ It's important to realize, as Bill Dubuque pointed out, that you must have $\gcd(B, n) = 1$. But if so, if $B \equiv b$ then $1\equiv B*B^{-1} \equiv b*B^{-1}$ and $b^{-1} \equiv b^{-1}(b*B^{-1}) \equiv B^{-1}$. Which isnt surprising as equivalence carries over multiplication and we are multiplying the inverses to get $1$ $\endgroup$ – fleablood Feb 12 at 18:01
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As $2$ is relatively prime to $125$ then $2$ is invertable so there is a $[\frac 12]$ so that $2[\frac 12]\equiv 1 \pmod {125}$ (just let $[\frac 12] = 63$ but we don't actually care what $[\frac 12]$ is; just that it exists) so for any $m = 2^jb$ then $\frac m{2^j}\equiv \frac m{2^j}(2^j*[\frac 12]^j) \equiv m*[\frac 12]^j$.

So $\frac {6^{666}}{2^6} \equiv 6^{666}[\frac 12]^6 \equiv 6^{\phi (666)}[\frac 12]^6\pmod {125}$

For notation purposes it is pefectly acceptable to write $\frac 12 \equiv 63 \pmod {125}$ and to use the fraction notation. (Although it's perhaps misleading to use the $2^{-1} \equiv 63 \pmod {125}$ notation instead.)

It's just important to realize that the residuce class is not $\{\frac 12 + 125k| k \in \mathbb Z\}$ but $\{m|2m \equiv 1 \pmod 125\} = \{m|\exists k\in \mathbb Z; 2m = 1 + 125k\}=\{m|2m = 1 + 125k$ for some odd $k\} =\{\frac {1+125(2k+1)}2| k \in \mathbb Z\}=\{63+ 125k|k \in \mathbb Z\}$.

And it's important to realize that if $k$ and $n$ arent relatively prime there isn't and such $k^{-1} \mod n$ and we can't use $\frac 1k \pmod n$

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