Show that ## 11\mid R_{n} ## if and only if ## n ## is even

[Math100]

Homework Statement

Given a repunit $R_{n}$, show that $11\mid R_{n}$ if and only if $n$ is even.

Relevant Equations

None.

Proof:

Suppose $11\mid R_{n}$, given a repunit $R_{n}$.
Let $R_{n}=1\cdot 10^{m}+\dotsb +1\cdot 10+1$ and $T=(a_{0}-a_{1})+(a_{2}-a_{3})+\dotsb +(-1)^{m}a_{m}$.
Then $T=(1-1)+(1-1)+\dotsb +(-1)^{m}a_{m}=0$.
This means $11\mid R_{n}\implies T=0$.
Thus, $n$ is even.
Conversely, suppose $n$ is even.
Then $1-1+1-1+\dotsb -1+1=0$ and $11\mid 0$.
Thus $11\mid R_{n}$.
Therefore, $11\mid R_{n}$ if and only if $n$ is even.

 

[fresh_42]

Homework Statement:: Given a repunit $R_{n}$, show that $11\mid R_{n}$ if and only if $n$ is even.
Relevant Equations:: None.

Proof:

Suppose $11\mid R_{n}$, given a repunit $R_{n}$.
Let $R_{n}=1\cdot 10^{m}+\dotsb +1\cdot 10+1$ and $T=(a_{0}-a_{1})+(a_{2}-a_{3})+\dotsb +(-1)^{m}a_{m}$.
Then $T=(1-1)+(1-1)+\dotsb +(-1)^{m}a_{m}=0$.
This means $11\mid R_{n}\implies T=0$.
Thus, $n$ is even.
Conversely, suppose $n$ is even.
Then $1-1+1-1+\dotsb -1+1=0$ and $11\mid 0$.
Thus $11\mid R_{n}$.
Therefore, $11\mid R_{n}$ if and only if $n$ is even.

The first part is a bit clumsy and a link to the thread with that criterion that you use would have been nice, meaning: I couldn't find it.

I would write the first part as:

Suppose $11\mid R_{m}$, given a repunit $R_{m}$.
Let $R_{m}=1\cdot 10^{m}+\dotsb +1\cdot 10+1$ and $T=(a_{0}-a_{1})+(a_{2}-a_{3})+\dotsb +(-1)^{m}a_{m}$.
Then (insert link) $T=0$ and $m$ has to be even.

The second part is also shown in
https://www.physicsforums.com/threa...-number-of-digits-is-divisible-by-11.1017287/

 

[Math100]

The first part is a bit clumsy and a link to the thread with that criterion that you use would have been nice, meaning: I couldn't find it.

I would write the first part as:

Suppose $11\mid R_{m}$, given a repunit $R_{m}$.
Let $R_{m}=1\cdot 10^{m}+\dotsb +1\cdot 10+1$ and $T=(a_{0}-a_{1})+(a_{2}-a_{3})+\dotsb +(-1)^{m}a_{m}$.
Then (insert link) $T=0$ and $m$ has to be even.

The second part is also shown in
https://www.physicsforums.com/threa...-number-of-digits-is-divisible-by-11.1017287/

But what specifically, should I insert in the first part of the proof, where "Then (insert link) $T=0$..."?

 

[fresh_42]

But what specifically, should I insert in the first part of the proof, where "Then (insert link) $T=0$..."?

Sorry, my fault. You just use the divisibility by 11 criterion. I thought it was an earlier thread. I haven't had it in mind. So maybe a reminder for people like me would be nice: A number is divisible by 11 if and only if its alternating sum of digits is.