Show: n!+1 & (n+1)!+1 are relatively prime

kingwinner

Prove that n! + 1 and  (n+1)! + 1 are relatively prime.
[note: greatest common divisor=gcd=highest common factor=hcf; a and b are relatively prime means that gcd(a,b)=1]

呢題我唔係好識點 prove...用 mathematical induction prove 可以嗎?

thx for any help

[ 本帖最後由 kingwinner 於 10-1-8 07:22 AM 編輯 ]

Naozumi

Suppose
Let a = n! +1, b = (n+1)! + 1.
(n+1)a - b = [(n+1)!  + (n+1)] - [(n+1)! + 1]
                 = n

But gcd(a, b) = gcd(n!+1, n)
Clearly, n! + 1 and n are relatively prime.
So.....

[ 本帖最後由 Naozumi 於 10-1-10 11:17 PM 編輯 ]

kingwinner

原帖由 Naozumi 於 10-1-10 08:54 PM 發表
Suppose
Let a = n! +1, b = (n+1)! + 1.
(n+1)a - b = [(n+1)!  + (n+1)] - [(n+1)! + 1]
                 = n

But gcd(a, b) = gcd(n!+1, n)
Clearly, n! + 1 and n are relatively prime.
So.....

我知道 n! and n!+1 are relatively prime (2 consecutive integers are always relatively prime), 但點 prove n! + 1 and n are relatively prime?

Naozumi

唉....
我當 n > 1
n! + 1 一定唔可以比2,3,....,n整除啦!!!
n的因子咪又係 2,3,...,n呢d數!

[ 本帖最後由 Naozumi 於 10-1-11 02:46 PM 編輯 ]