- #1

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- 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.

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.