<s>[INST] Prove that $\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}$ is an integer. Prove that $$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}$$ is an integer using mathematical induction.
I tried using mathematical induction but using binomial formula also it becomes little bit complicated.
Please show me your proof.
Sorry if this question was already asked. Actually i did not found it. In that case only sharing the link will be enough.
 [/INST] Base case for $k=1$: $$\frac{1^7}{7}+\frac{1^5}{5}+\frac{2*1^3}{3}-\frac{1}{105}=1$$
Now, assume for some k that $\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}$ is indeed in integer.  
Then $$\frac{(k+1)^7}{7}+\frac{(k+1)^5}{5}+\frac{2(k+1)^3}{3}-\frac{k+1}{105}=\\\frac{\sum_{i=0}^7\binom{7}{i}k^i}{7}+\frac{\sum_{i=0}^5\binom{5}{i}k^i}{5}+2\frac{\sum_{i=0}^3\binom{3}{i}k^i}{3}-\frac{k+1}{105}=$$ 
Extracting the highest indexed term from each sum (and the $-\frac{k}{105}$ at the end):  $$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}+\frac{\sum_{i=0}^6\binom{7}{i}k^i}{7}+\frac{\sum_{i=0}^4\binom{5}{i}k^i}{5}+2\frac{\sum_{i=0}^2\binom{3}{i}k^i}{3}-\frac{1}{105}$$ 
By the induction hypothesis, the sum of the first four terms is an integer so, if we can show the rest of the above sum is an integer, we will be done. Use the fact that, for any prime, $p$, $p|\binom{p}{k}$ where $1\leq k\leq p-1$. This is because $$\binom{p}{k}=\frac{p(p-1)...(p-k+1)}{k(k-1)...1}$$ 
$p$ divides the numerator but not the denominator (as $1\leq k\leq p-1$) so $p|\binom{p}{k}$ 
So each term in the remaining sum with index $i$ $\geq1$ and $\leq p-1$ ($p$ being the respective prime in each sum) is divisible by the corresponding $p$ in the denominator and produces an integer. The only non-integer terms left will be the ones at $i=0$, i.e.  $$\frac{\binom{7}{0}k^0}{7}+\frac{\binom{5}{0}k^0}{5}+\frac{2\binom{3}{0}k^0}{3}-\frac{1}{105}=\frac{1}{7}+\frac{1}{5}+\frac{2}{3}-\frac{1}{105}=1$$ 
So $\frac{(k+1)^7}{7}+\frac{(k+1)^5}{5}+\frac{2(k+1)^3}{3}-\frac{k+1}{105}$ is a sum of integers making it an integer.
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