JEE Advance - Mathematics (1990 - No. 13)
Prove that $${{{n^7}} \over 7} + {{{n^5}} \over 5} + {{2{n^3}} \over 3} - {n \over {105}}$$ is an integer for every positive integer $$n$$
The expression is always an integer due to Fermat's Little Theorem and modular arithmetic.
The expression is an integer if and only if n is a multiple of 7.
The expression is an integer only for prime values of n.
The expression is an integer if and only if n is odd.
The expression is not always an integer; there exist positive integers n for which it is not an integer.