## anonymous 4 years ago First answer (by Anders) on http://www.quora.com/Mathematics/How-would-I-show-that-49-divides-8-n-7n-1-for-all-n-ge-0 Can anyone explain it to me? Hints'd be appreciated.

1. anonymous

@mukushla @KingGeorge @Zarkon

2. anonymous

can u type the equation here plz :)

3. anonymous

@mukushla try again.

4. anonymous

I am not concerned with problem as much as its solution on Quora. And No @nincompoop

5. anonymous

thats not a true statement of binomial theorem$8^n=(1+7)^n=\sum_{k=0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right) 7^n$right?

6. anonymous

I am pretty sure, it is.

7. anonymous

oops sorry that should be$8^n=(1+7)^n=\sum_{k=0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right) 7^k$

8. anonymous

oh yeah. k. lol

9. anonymous

but that's not what concerns me. it's the first answer by Anders i am unable to understand.

10. anonymous

oh sorry man lol :) im sooo blind

11. anonymous

i cant understand it :)

12. experimentX

$8^n=(1+7)^n=\sum_{k=0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right) 7^n = 1 + 7n + O(7^{(n \geq 2)})$

13. anonymous

Sai Ganesh explanation also no good, right?

14. experimentX

$$O(7^{(n \geq 2)})$$ is always divisible by 49

15. anonymous

Yes, sorry, I was directing at Ishaan (he wants an "octal" explanation)

16. anonymous

Yeah. It didn't help my puny brain much :(

17. anonymous

@him1618 maybe you can?

18. anonymous

the explanation is good enough

19. anonymous

I have to say that the binomial answer is much more natural, it would not occur to me to set out on an octal adventure for that particular problem....

20. anonymous

true

21. anonymous

Specifically, I don't understand how did he conclude 'The rest can be divided into equal groups based on their first two nonzero digits.'

22. anonymous

... 49* equal groups ...

23. anonymous

7*7

24. anonymous

you got rid of the 0's so you got 7 digits left you can set, right?

25. anonymous

right.

26. anonymous

So if all the numbers are divisible into 49 groups, then divisibilty by 49 follows. At least, that's what I think it says.

27. anonymous

Okay, one silly doubt. Why is he using only first two digits for grouping?

28. anonymous

because 7*7 = 49

29. anonymous

i was acting foolishly yesterday :( thanks.