Question

Old problem: Show that if n is odd or a multiple of 4, then 1^3 +2^3 +...+(n-1)^3 equivalent 0 (mod n)
Please, help

Answer 1

I got stuck at the last argument.

Answer 2

\(\sum_{i=1}^n i^3 =\dfrac{1}{4}n^2(n+1)^2\)
Hence \(\sum_{i=1}^{n-1}i^3 = \sum_{i=1}^n i^3 - n^3\)
but \(n|n^3\) , hence
\(\sum_{i=1}^{n-1}i^3\equiv \sum_{i=1}^n i^3\)
Hence we need find the condition of n, such that \(\dfrac {1}{4}n^2 (n+1)^2\) divisible by n

Answer 3

We have 4 = 2^2, hence \(n\equiv 0,1\) (mod2)
If \(n\equiv 0(mod2)\) , then the whole thing divisible by 4
If \(n\equiv 1(mod2)\), then \((n+1) \equiv 0(mod2)\) and the whole thing divisible by n also.

Answer 4

But then, I just have the condition that n is divisible by 2 only, not 4 as require. What is wrong with my argument?

Answer 5

@ikram002p

Answer 6

.

Answer 7

(n(n+1)/2)^2 must be divisible by n so simplifing the expression, n(n+1)^2/4 must be an integer.

That means either n/4 is an integer; i.e. n is a multiple of 4; or
(n+1)^2/4 is an integer; i.e. n must be odd so that n+1 is even and (n+1)^2 will be divisible by 4.

Answer 8

SPAM

Answer 9

Question: if n is even, we have \(\dfrac{1}{4} n*n (n+1)^4\)
the first n divisible by2, same as the second one. Why do we need 4 | n?

Answer 10

if n is odd, n+1 is even and same argument, we have (n+1) (n+1) on the expression. Hence for all n odd, the expression divisible by 4 and that makes it be integer. Am I right?

Answer 11

oh, I got it.

Answer 12

after dividing by n, the expression becomes (n*(n+1)^2)/4 and that needs to be an integer for the sum to be divisible by 4.

....not sure where the (n+1)^4 came from...

Answer 13

oh cool :)

Answer 14

the first n must divisible by 4 to cancel the 4 at the bottom. And then the second one make the whole divisible by n.

Answer 15

for (n+1)^4, it is a mistake, it should be (n+1)^2 :) sorry.

Answer 16

Either the first term, n, must be divisible by 4 to cancel the 4 at the bottom; OR n must be odd to make the second term divisible by 4. Then the summation is divisible by n.

Answer 17

Thank you so much. I do appreciate

Answer 18

welcome n excuse me 4 not typing things out in LaTex - i dont have much luck w it on my computer