## Ishaan94 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. one year ago one year ago

1. Ishaan94

@mukushla @KingGeorge @Zarkon

2. mukushla

can u type the equation here plz :)

3. Ishaan94

@mukushla try again.

4. Ishaan94

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

5. mukushla

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. Ishaan94

I am pretty sure, it is.

7. mukushla

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. Ishaan94

oh yeah. k. lol

9. Ishaan94

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

10. mukushla

oh sorry man lol :) im sooo blind

11. mukushla

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. estudier

Sai Ganesh explanation also no good, right?

14. experimentX

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

15. estudier

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

16. Ishaan94

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

17. Ishaan94

@him1618 maybe you can?

18. him1618

the explanation is good enough

19. estudier

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. him1618

true

21. Ishaan94

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. Ishaan94

... 49* equal groups ...

23. estudier

7*7

24. estudier

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

25. Ishaan94

right.

26. estudier

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. Ishaan94

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

28. estudier

because 7*7 = 49

29. Ishaan94

i was acting foolishly yesterday :( thanks.