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# Sum of odd powers of n consecutive numbers (TOMATO subj 31)

If k is an odd positive integer, prove that for any integer $$\mathbf{ n ge 1 , 1^k + 2^k + \cdots + n^k }$$ is divisible by $$\mathbf{ \frac {n(n+1)}{2} }$$

Discussion:
We write the given expression in two ways:

$$\mathbf{ S= 1^k + 2^k + \cdots + n^k }$$
$$\mathbf{ S =n^k + (n-1)^k + \cdots + 1^k }$$