Answer to Question #326605 in Combinatorics | Number Theory for Allen

Question

Prove: 12|N iff 4| N ^ 3 | N.

Answer 1

$Let\,\,12|N\Rightarrow N=12k\Rightarrow N=3\left( 4k \right) ,N=4\left( 3k \right) \Rightarrow 4|N\land 3|N\\Let\,\,4|N\land 3|N\Rightarrow \left\{ \begin{array}{c} N=4k\\ N=3m\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} N=4k\\ 4k=3m\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} N=4k\\ k=3\left( m-k \right)\\\end{array} \right. \Rightarrow \\\Rightarrow N=4\cdot 3\left( m-k \right) \Rightarrow N=12\left( m-k \right) \Rightarrow 12|N$