Prove: 12|N iff 4| N ^ 3 | N.
Let 12∣N⇒N=12k⇒N=3(4k),N=4(3k)⇒4∣N∧3∣NLet 4∣N∧3∣N⇒{N=4kN=3m⇒{N=4k4k=3m⇒{N=4kk=3(m−k)⇒⇒N=4⋅3(m−k)⇒N=12(m−k)⇒12∣NLet\,\,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|NLet12∣N⇒N=12k⇒N=3(4k),N=4(3k)⇒4∣N∧3∣NLet4∣N∧3∣N⇒{N=4kN=3m⇒{N=4k4k=3m⇒{N=4kk=3(m−k)⇒⇒N=4⋅3(m−k)⇒N=12(m−k)⇒12∣N
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