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 $\emptyset$