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How do I prove the 11 modulo rule?

 
 
TangDavid的头像
How do I prove the 11 modulo rule?
TangDavid - 2020年03月22日 Sunday 18:24
 

Title says all!

-DT

 
ReynosoDavid的头像
Re: How do I prove the 11 modulo rule?
ReynosoDavid - 2020年03月23日 Monday 09:05
 

Notice that $10 \equiv -1 \pmod{11}$. Thus $$10^n \equiv \begin{cases}1 & n \text{ even} \\ -1 &  n \text{ odd}\end{cases} \pmod{11}$$ Then use this fact to prove that a number is congruent to the alternating sum of its digits $\pmod{11}$.