Question Number 188156 by Shrinava last updated on 26/Feb/23
$$\mathrm{a},\mathrm{b},\mathrm{c}\in\mathbb{N} \\ $$$$\mathrm{5a}\:=\:\mathrm{6b}\:=\:\mathrm{9c} \\ $$$$\left(\mathrm{abc}\right)_{\boldsymbol{\mathrm{min}}} \:=\:? \\ $$
Answered by mr W last updated on 26/Feb/23
![5a = 6b = 9c=n 5a = 2×3b = 3^2 c=n ⇒n_(min) =5×2×3^2 =90 [or n_(min) =lcm(5,6,9)=90] ⇒a_(min) =((90)/5)=18 ⇒b_(min) =((90)/6)=15 ⇒c_(min) =((90)/9)=10 ⇒(abc)_(min) =18×15×10=2700 ✓](Q188160.png)
$$\mathrm{5a}\:=\:\mathrm{6b}\:=\:\mathrm{9c}={n} \\ $$$$\mathrm{5a}\:=\:\mathrm{2}×\mathrm{3b}\:=\:\mathrm{3}^{\mathrm{2}} \mathrm{c}={n} \\ $$$$\Rightarrow{n}_{{min}} =\mathrm{5}×\mathrm{2}×\mathrm{3}^{\mathrm{2}} =\mathrm{90} \\ $$$$\left[{or}\:{n}_{{min}} ={lcm}\left(\mathrm{5},\mathrm{6},\mathrm{9}\right)=\mathrm{90}\right] \\ $$$$\Rightarrow{a}_{{min}} =\frac{\mathrm{90}}{\mathrm{5}}=\mathrm{18} \\ $$$$\Rightarrow{b}_{{min}} =\frac{\mathrm{90}}{\mathrm{6}}=\mathrm{15} \\ $$$$\Rightarrow{c}_{{min}} =\frac{\mathrm{90}}{\mathrm{9}}=\mathrm{10} \\ $$$$\Rightarrow\left({abc}\right)_{{min}} =\mathrm{18}×\mathrm{15}×\mathrm{10}=\mathrm{2700}\:\checkmark \\ $$
Commented by Shrinava last updated on 26/Feb/23
$$\mathrm{cool}\:\mathrm{dear}\:\mathrm{professor}\:\mathrm{thankyou} \\ $$