hallelujah1234 wrote:

Is (n^5/5)+(n^3/3)+(7n/15) an integer?

1. n is odd

2. n is even

(3*n^5 + 5*n^3 + 7*n)/15 = n(3n^4+5*n^2 + 7)/15.

3n^4+5n^2+7 = (-2n^4+2)+5(n^4+n^2+1).

by the theorem of Ferma: n(n^4-1) :: 5, and by the theorem of Ferma n(n^2-1) :: 3. => first part is interger anyway.

Second part: n(n^4+n^2+1)=n([n^4-n^2]+[2n^2+1]). Again, by the theorem of Ferma, n(n^4-n^2)=n(n^2-1)*(n^2) :: 3. n(2n^2+1) = n(2n^2-2+3) :: 3 again be the theorem of Ferma!

Regardless of 1 or 2 the stem is true!

P.S. theorem of Ferma: n^p-n always :: p, where p is prime, n - integer.