Find the remainder when ## 823^{823} ## is divided by ## 11 ##

[Math100]

Homework Statement

Find the remainder when ## 823^{823} ## is divided by ## 11 ##.

Relevant Equations

Euler-Fermat theorem.
Assume ## (a, m)=1 ##.
Then we have ## a^{\phi(m)}\equiv 1\pmod {m} ##.

Observe that ## 823\equiv 9\pmod {11}\equiv -2\pmod {11} ##.
This implies ## 823^{823}\equiv (-2)^{823}\pmod {11} ##.
Applying the Euler-Fermat theorem produces:
## gcd(-2, 11)=1 ## and ## (-2)^{\phi(11)}\equiv 1\pmod {11} ##.
Since ## \phi(p)=p-1 ## where ## p ## is any prime, it follows that ## \phi(11)=10 ##.
Now we have ## (-2)^{10}\equiv 1\pmod {11} ##.
Thus
\begin{align*}
&(-2)^{823}\equiv [((-2)^{10})^{82}\cdot (-2)^3]\pmod {11}\\
&\equiv [(1)^{82}\cdot (-8)]\pmod {11}\\
&\equiv -8\pmod {11}\\
&\equiv 3\pmod {11}.\\
\end{align*}
Therefore, ## 823^{823}\equiv 3\pmod {11} ## and the remainder when ## 823^{823} ## is divided by ## 11 ## is ## 3 ##.

 

[fresh_42]

Homework Statement:: Find the remainder when ## 823^{823} ## is divided by ## 11 ##.
Relevant Equations:: Euler-Fermat theorem.
Assume ## (a, m)=1 ##.
Then we have ## a^{\phi(m)}\equiv 1\pmod {m} ##.

Observe that ## 823\equiv 9\pmod {11}\equiv -2\pmod {11} ##.
This implies ## 823^{823}\equiv (-2)^{823}\pmod {11} ##.
Applying the Euler-Fermat theorem produces:
## gcd(-2, 11)=1 ## and ## (-2)^{\phi(11)}\equiv 1\pmod {11} ##.
Since ## \phi(p)=p-1 ## where ## p ## is any prime, it follows that ## \phi(11)=10 ##.
Now we have ## (-2)^{10}\equiv 1\pmod {11} ##.
Thus
\begin{align*}
&(-2)^{823}\equiv [((-2)^{10})^{82}\cdot (-2)^3]\pmod {11}\\
&\equiv [(1)^{82}\cdot (-8)]\pmod {11}\\
&\equiv -8\pmod {11}\\
&\equiv 3\pmod {11}.\\
\end{align*}
Therefore, ## 823^{823}\equiv 3\pmod {11} ## and the remainder when ## 823^{823} ## is divided by ## 11 ## is ## 3 ##.

Correct.

Just a remark: Euler's function is usually written varphi ##\varphi ##

 

[Math100]

Correct.

Just a remark: Euler's function is usually written varphi ##\varphi ##

So ## \varphi ## is different from ## \phi ##?

 

[fresh_42]

So ## \varphi ## is different from ## \phi ##?

##\phi## is the capital "F" and ##\varphi ## the lower case "f".
That Tex also allows ##\Phi## doesn't change this fact. This is more of the distinction between "F" and "F".

 

[haruspex]

##\phi## is the capital "F"

I don't think that's right. The italicisation makes it lower case. See e.g. https://office-watch.com/2021/how-to-type-phi-uppercase-φ-or-lower-φ-in word-and-office/

 

[fresh_42]

I don't think that's right. The italicisation makes it lower case. See e.g. https://office-watch.com/2021/how-to-type-phi-uppercase-φ-or-lower-φ-in word-and-office/

I do not care any artificial settings. I think only Greek counts:
https://en.wikipedia.org/wiki/Greek_alphabet

 

[haruspex]

I do not care any artificial settings. I think only Greek counts:
https://en.wikipedia.org/wiki/Greek_alphabet

Ok, that source does not use italics to distinguish (though another wikipedia link does show that distinction), but neither does it support your view that ##\phi## is uppercase. Rather, it uses the rather subtle variation in the vertical position of the character and the slightly less subtle variation in tail length. Both confirm my view that ##\phi## is lowercase.
I would guess that the distinction between ##\phi## and ##\varphi## is like that between block letters and cursive.
Note the corresponding trio of forms ##\theta, \vartheta, \Theta## at https://www.overleaf.com/learn/latex/List_of_Greek_letters_and_math_symbols.

 

[fresh_42]

There is only one "f" in the Greek alphabet, ##\varphi ## for the lower and ##\phi## for the upper case. Everything else is an invention of the 20th century and irrelevant to my argumentation.

The Cyrillic alphabet writes "f" and ##\phi## and ##\Phi .## And although Euler worked at the Tsar's court, he had chosen the Greek ##\varphi ## for the totient function, and not the Russian ##\phi.##

 

[haruspex]

Everything else is an invention of the 20th century and irrelevant to my argumentation.

That does not fit with your position in post #4, where you state that ##\phi## corresponds to F. Besides, what matters to mathematicians now is current standard usage by other mathematicians.
I would not care about all this except that I fear you are misleading @Math100 .

 

[fresh_42]

That does not fit with your position in post #4, where you state that ##\phi## corresponds to F. Besides, what matters to mathematicians now is current standard usage by other mathematicians.
I would not care about all this except that I fear you are misleading @Math100 .

Misleading would be to allow Euler's function to be named by ##\phi.##

 

[haruspex]

Misleading would be to allow Euler's function to be named by ##\phi.##

As at https://brilliant.org/wiki/eulers-totient-function/, https://www.whitman.edu/mathematics/higher_math_online/section03.08.html, https://mathworld.wolfram.com/TotientFunction.html, , https://www.geeksforgeeks.org/eulers-totient-function/, https://cp-algorithms.com/algebra/phi-function.html, https://artofproblemsolving.com/wiki/index.php/Euler's_totient_function, https://arxiv.org/abs/2110.09875, https://math.stackexchange.com/ques...t-function-of-a-product-for-arbitrary-n-and-m …?

As far as I was aware and discern from all these examples, the distinction between ##\phi## and ##\varphi## is an arbitrary choice of font. This is also supported at https://math.stackexchange.com/questions/1411557/notation-varphi-and-phi and https://en.wikipedia.org/wiki/Phi, which states:
"The lowercase letter φ (or often its variant, ϕ) is often used to represent the following:
…
Euler's totient function φ(n) in number theory".

Somewhere in all that I read that ##\phi, \theta, …## is favoured in the US, ##\varphi, \vartheta, ..## in Europe (but that's news to me).

This would seem to be a long-standing misunderstanding on your part. Welcome to the club.

 

[fresh_42]

As at https://brilliant.org/wiki/eulers-totient-function/, https://www.whitman.edu/mathematics/higher_math_online/section03.08.html, https://mathworld.wolfram.com/TotientFunction.html, , https://www.geeksforgeeks.org/eulers-totient-function/, https://cp-algorithms.com/algebra/phi-function.html, https://artofproblemsolving.com/wiki/index.php/Euler's_totient_function, https://arxiv.org/abs/2110.09875, https://math.stackexchange.com/ques...t-function-of-a-product-for-arbitrary-n-and-m …?

As far as I was aware and discern from all these examples, the distinction between ##\phi## and ##\varphi## is an arbitrary choice of font. This is also supported at https://math.stackexchange.com/questions/1411557/notation-varphi-and-phi and https://en.wikipedia.org/wiki/Phi, which states:
"The lowercase letter φ (or often its variant, ϕ) is often used to represent the following:
…
Euler's totient function φ(n) in number theory".

Somewhere in all that I read that ##\phi, \theta, …## is favoured in the US, ##\varphi, \vartheta, ..## in Europe (but that's news to me).

This would seem to be a long-standing misunderstanding on your part. Welcome to the club.

I consider such usage as wrong for logical reasons deduced from the Greek alphabet. Wrong doesn't become right by numbers. The lower case Greek "f" is "##\varphi ##". ##\phi## is an uppercase letter. I do not see how this could be disputable. Since when do modern attitudes change ancient facts?

I am willing to discuss whether it is fundamentally reasonable to have naming conventions at all, but that would be a different topic.

 

[TeethWhitener]

This discussion is off topic but I have to agree with @haruspex, and apparently @fresh_42 agrees with him too:

I do not care any artificial settings. I think only Greek counts:
https://en.wikipedia.org/wiki/Greek_alphabet

The wiki link clearly gives ##\Phi## as uppercase and ##\phi## or ##\varphi## as lowercase (see heading “glyph variants”), as does the wiki article on the letter itself:
https://en.m.wikipedia.org/wiki/Phi

Edit: just to hammer the last nail in this coffin, the Greek Wikipedia page for phi confirms:
https://el.m.wikipedia.org/wiki/Φι

 

[fresh_42]

We have three symbols and only two "f". Now does ##\phi## looks more like ##\Phi## or more like ##\varphi ##?

 

[TeethWhitener]

It looks more like ##\emptyset##