proof using the mathematical induction Prove $$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}$$ is an integer for every positive integer k.
In proving for (n+1) integer,the expression is integer,I found $(n+1)^7$ term.I use binomial theorem to expand but finally it won't work(some term become integer While some other remaining as rational)
Any hint to solve the problem is appreciated.Thanks.

We have, for instance, $(k+1)^7/7 = (\sum_{i=0}^7 \binom{7}{i}k^i)/7$. Note that the only times the coefficient of $k^i$ in the numerator is not divisible by $7$ is when $i=0,7$. Therefore, $(k+1)^7/7 = k^7/7 + 1/7 + \text{integer}$. Applying the same logic to the other terms, we find
$$\begin{align}\frac{(k+1)^7}{7}+\frac{(k+1)^5}{5}+\frac{2(k+1)^3}{3}-\frac{k+1}{105}&=\left(\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}\right) \\&+\left(\frac{1}{7}+\frac{1}{5}+\frac{2}{3}-\frac{1}{105}\right) + \text{integer}\\ &= \text{integer} + 1 +\text{integer} \\&= \text{integer}.\end{align}$$
