Note

Number Theory Level 3

I found the note below in my pocket.

What follows from this?

A : If $k$ is a positive integer, then $k\mid k^3-k$ .

B : If $k$ is a positive integer, then $(1+2+3+\dots+k)^2=1^2+2^3+3^3+\dots+k^3$ .

C : If $k$ is a positive integer, then $k\times (k+1)\times (k+2)\times (k+3) +1$ is a perfect square.

None of them B C A

1 solution

Kenny O.
Aug 21, 2017

let n+1=k
(k-1)(k)(k+1)+k=k(k^2 -1) +k = k^3-k +k =k^3 k(k-1)(k+1)= k^3- k
Thus, $k| k^3+k$ is the answer.
B and C are not answers as any $a^2$ is not in the note (other than 64).

Note

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