Prove that

Printable View

- Aug 17th 2010, 07:20 AMelimA divisibility problem
Prove that

- Aug 18th 2010, 12:18 AMaman_cc
Hint - Consider/compare the power of any prime, p, in the expansion of and

- Aug 18th 2010, 01:25 AMsimplependulum
Let and be the max. power of prime dividing and respectively , then we need to show that

for any

First note that so

For , we thus have

Let and , then

( since if and now , . )

Obviously ,

Therefore ,

We can prove by induction , :

It is true when and assume the case is true for any natural number . When we jump to the next case , we have which completes the proof . - Aug 18th 2010, 04:16 AMaman_cc
It may be enough to note/prove that for any n,m,a (all positive integers).

This can be shown easily using induction on m.

Once this is established use this fact in the series to find the power of a prime in n! i.e. N_p(n) (as per notation used by simplependulum)

This might just give a little simpler proof for the same. - Aug 18th 2010, 05:43 AMsimplependulum
It is false when , perhaps we need to elaborate more at this point . That's why i need to introduce the substitution .

- Aug 18th 2010, 05:43 AMsimplependulum
Edited .

- Aug 19th 2010, 08:21 PMaman_cc
Thanks. Let me re-check my work please.