Question

Does n/2 divides n!
Please, explain

Answer 1

is n even ?

Answer 2

yes

Answer 3

then yes because
1,2,3,...n/2,....n

all of them divide n!

Answer 4

n! = 1.2.3....n/2...n

Answer 5

Thank you!!

Answer 6

let me rearrange my question :)

Answer 7

okay :)
it is easy to see that `n!` is divisible by all numbers `<= n`

Answer 8

*all positive integers

Answer 9

if n =9, then 4*5 divides 9!?

Answer 10

hahaha... my stupid question, of course it is ...

Answer 11

but how to arrange it in term of n??

Answer 12

I mean 4 = (n-1)/2
5 = (n+1) /2

Answer 13

even numbers : 2k
odd numbers : 2k-1

Answer 14

if n is odd, (n-1) would be even, yes ?

Answer 15

yes

Answer 16

you want to ask :

Does (n-1)/2 divide n! when n is odd ?

Answer 17

Ok, let me retype it: Let S_n is a sequence

If n is even, then Sn can be broken into 2-equal- parts, each of them has n/2 term
I have to prove that l.c.m ( n/2, n/2) divides n!. The case close by your help.
If n is odd, then Sn can be broken into 2 parts also, the first part has (n-1)/2 terms, the second part has (n+1)/2 terms. I have to prove
\(l.c.m~~ (\dfrac{n-1}{2}, \dfrac{n+1}{2})\) divides n!

Answer 18

Surely, one is even and one is odd, so that \(l.c.m= (\dfrac{n-1}{2})( \dfrac{n+1}{2})\)

Answer 19

@rational I am sorry, the net is so bad here....

Answer 20

when n is odd :

n! = 1.2.3...(n-1)/2.(n+1)/2....n

Answer 21

(n-1)/2 and (n+1)/2 are two DIFFERENT numbers less than n,
so their product divides n! for obvious reasons ?

Answer 22

Yes, I got it. Thank you so much.