Show that if $k$ is odd,

\[ 1+2+\cdots+n \]

divides

\[1^k +2^k +\cdots+n^k\]

## Sum of integers divide sum of integer powers

### Sum of integers divide sum of integer powers

"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

Please

Please

**install LaTeX fonts**in your PC for better looking equations,**learn****how to write equations**, and**don't forget**to read Forum Guide and Rules.### Re: Sum of integers divide sum of integer powers

$1+2+....+n=\frac {n(n+1)} 2$

It is well known that $a+b|a^k+b^k$ for $k$ odd.

Using this $n+1|n^k+1^k,(n-1)^k+2^k,......$

$n|n^k,(n-1)^k+1^k,...$ and $gcd(n,n+1)=1$,so if both of them divides this then there product divide $2(1^k+....+n^k)$

It is well known that $a+b|a^k+b^k$ for $k$ odd.

Using this $n+1|n^k+1^k,(n-1)^k+2^k,......$

$n|n^k,(n-1)^k+1^k,...$ and $gcd(n,n+1)=1$,so if both of them divides this then there product divide $2(1^k+....+n^k)$

One one thing is neutral in the universe, that is $0$.

### Re: Sum of integers divide sum of integer powers

I am posting a very easy solution....it can't be accepted by all....

we know that if k is odd then ...

a^k + b^k/a + b

so why we can't say that

a^k + b^k + ............. +n^k/a + b +.............+n [a,b belong to N and k is an odd]

we know that if k is odd then ...

a^k + b^k/a + b

so why we can't say that

a^k + b^k + ............. +n^k/a + b +.............+n [a,b belong to N and k is an odd]

### Re: Sum of integers divide sum of integer powers

I don't think it is true.How did you get this?

One one thing is neutral in the universe, that is $0$.