Question #326605

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


1
Expert's answer
2022-04-13T18:10:32-0400

Let12NN=12kN=3(4k),N=4(3k)4N3NLet4N3N{N=4kN=3m{N=4k4k=3m{N=4kk=3(mk)N=43(mk)N=12(mk)12NLet\,\,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


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