Question Number 1776 by 123456 last updated on 20/Sep/15
$$\boldsymbol{\mathrm{P}}_{{k}} =\left\{{x}\in\mathbb{N},{n}\in\mathbb{N}:{x}>\mathrm{0},\underset{{n}\mid{x}} {\sum}\mathrm{1}={k}\right\} \\ $$$$\boldsymbol{\mathrm{C}}=\underset{{k}\geqslant\mathrm{3}} {\cup}\boldsymbol{\mathrm{P}}_{{k}} \\ $$$$\left\{\mathrm{0},\mathrm{1}\right\}\cup\boldsymbol{\mathrm{C}}\cup\mathbb{P}=\mathbb{N} \\ $$$$\mathrm{proof}\:\mathrm{or}\:\mathrm{give}\:\mathrm{a}\:\mathrm{counter}\:\mathrm{example}\:\mathrm{that} \\ $$$$\left({x},{y},{z}\right)\in\boldsymbol{\mathrm{C}}^{\mathrm{3}} ,{x}^{\mathrm{2}} \mid{yz}\Rightarrow{x}\mid{y}\vee{x}\mid{z} \\ $$