Show that if n is a positive integer then the product n 1n h

Show that if n is a positive integer, then the product (n -1)n has the form 3k or 3k - 1 for some integer k >=0.

Solution

Case 1. n=3m for some integer m

(n-1)n=(3m-1)3m

Hence of the form 3k where ,k=m(3m-1)

Case 2. n=3m+1 for some integer m

(n-1)n=3m(3m+1)

Hence of the form 3k where,k=m(3m+1)

Case 3. n=3m+2 for some integer m

(n-1)n=(3m+1)(3m+2)=9m^2+6m+2=9m^2+6m+3-1

Hence of the form 3k-1

Hence proved.