Question Number 222736 by Lekhraj last updated on 06/Jul/25
Answered by Raphael254 last updated on 08/Sep/25
$$\begin{array}{|c|c|}{\boldsymbol{{To}}\:\boldsymbol{{a}}\:\boldsymbol{{number}}\:\boldsymbol{{be}}\:\boldsymbol{{divisible}}\:\boldsymbol{{by}}\:\mathrm{33},}\\{\boldsymbol{{it}}\:\boldsymbol{{needs}}\:\boldsymbol{{to}}\:\boldsymbol{{be}}\:\boldsymbol{{divisible}}\:\boldsymbol{{by}}\:\mathrm{3}\:\boldsymbol{{and}}\:\mathrm{11}}\\\hline\end{array} \\ $$$$ \\ $$$${Why}? \\ $$$$ \\ $$$${ab}\mid{n}\:\Rightarrow\:{a}\mid{n}\:{and}\:{b}\mid{n} \\ $$$$ \\ $$$$\begin{array}{|c|}{{if}\:{ab}\mid{n}:}\\\hline\end{array} \\ $$$$ \\ $$$${n}\:=\:{abp} \\ $$$$ \\ $$$${n}\:=\:\left({a}\right){bp} \\ $$$${n}\:=\:\left({b}\right){ap} \\ $$$$ \\ $$$${a}\mid{n}\:{and}\:{b}\mid{n} \\ $$$$ \\ $$$$\begin{array}{|c|}{{numerically}:}\\\hline\end{array} \\ $$$$ \\ $$$${a}\:=\:\mathrm{3},\:{b}\:=\:\mathrm{11},\:{n}\:=\:\mathrm{33}{r},\:{r}\:\in\:\mathbb{Z} \\ $$$$ \\ $$$$\mathrm{3}×\mathrm{11}\mid\mathrm{33}{r} \\ $$$$ \\ $$$$\mathrm{33}{r}\:=\:\mathrm{3}×\mathrm{11}×{p} \\ $$$$ \\ $$$$\mathrm{33}{r}\:=\:\left(\mathrm{3}\right)×\mathrm{11}×{p} \\ $$$$\mathrm{33}{r}\:=\:\left(\mathrm{11}\right)×\mathrm{3}×{p} \\ $$$$ \\ $$$$\mathrm{3}\mid\mathrm{33} \\ $$$$\mathrm{11}\mid\mathrm{33} \\ $$$$ \\ $$$$\:\underbrace{ } \\ $$