Prep T2 DS 73

[chris8888]

If n is a positive integer, is n3 - n divisible by 4 ?

(1) n = 2k + 1, where k is an integer.

(2) n2 + n is divisible by 6.

1). n^3 - n => n(n^2 - 1)
(2k+1)((2k+1)^2 - 1) => (2k+1)(4k^2 + 4k) 可以被4除

2). n(n + 1) / 2*3 and n(n^2 - 1)
不知道怎樣列式子算出, 我做對但是是憑感覺的.

[Huang Hsin-Yi]

n^3-n=4a ??
n(n^2-1)=n(n+1)(n-1)=4a ??

(2)n^2+n=6b
n(n+1)=6b代入上式
　6b(n-1)=4a
when b(n-1)=1 時 不能被4 除
　　　b(n-1)=4時　可被4除
therefore-----------------------------------insufficient