Question Number 116658 by zakirullah last updated on 05/Oct/20
Commented by Rasheed.Sindhi last updated on 08/Oct/20
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathbb{A}\mathrm{n}\mathbb{O}\mathrm{ther}\:\mathbb{W}\mathrm{ay}\:\curvearrowleft\underset{\downarrow} {\uparrow}\curvearrowright \\ $$$$\mathrm{12}=\mathrm{2}^{\mathrm{2}} .\mathrm{3} \\ $$$${The}\:{smallest}\:{perfect}\:{square}\: \\ $$$${divisible}\:{by}\:\mathrm{12}\::\:\mathrm{2}^{\mathrm{2}} .\mathrm{3}.\mathrm{3}=\mathrm{36} \\ $$$$\mathrm{16}=\mathrm{2}^{\mathrm{4}} \\ $$$${The}\:{smallest}\:{perfect}\:{square}\: \\ $$$${divisible}\:{by}\:\mathrm{16}\::\:\mathrm{2}^{\mathrm{4}} =\mathrm{16} \\ $$$$\mathrm{20}=\mathrm{2}^{\mathrm{2}} .\mathrm{5} \\ $$$${The}\:{smallest}\:{perfect}\:{square}\: \\ $$$${divisible}\:{by}\:\mathrm{20}\::\:\mathrm{2}^{\mathrm{2}} .\mathrm{5}.\mathrm{5}=\mathrm{100} \\ $$$$\mathrm{24}=\mathrm{2}^{\mathrm{3}} .\mathrm{3} \\ $$$${The}\:{smallest}\:{perfect}\:{square}\: \\ $$$${divisible}\:{by}\:\mathrm{24}\::\:\mathrm{2}^{\mathrm{3}} .\mathrm{3}.\mathrm{2}.\mathrm{3}=\mathrm{144} \\ $$$${The}\:{smallest}\:{perfect}\:{square} \\ $$$${number}\:{which}\:{is}\:{divisible} \\ $$$${by}\:\mathrm{12},\mathrm{16},\mathrm{20}\:\&\:\mathrm{24}: \\ $$$$\mathrm{LCM}\left(\mathrm{36},\mathrm{16},\mathrm{100},\mathrm{144}\right)=\mathrm{3600} \\ $$
Commented by zakirullah last updated on 08/Oct/20
$$\boldsymbol{{thank}}\:\boldsymbol{{so}}\:\boldsymbol{{much}}\:\boldsymbol{{sir}} \\ $$
Answered by mr W last updated on 05/Oct/20
$$\mathrm{12}=\mathrm{2}^{\mathrm{2}} ×\mathrm{3} \\ $$$$\mathrm{16}=\mathrm{2}^{\mathrm{4}} \\ $$$$\mathrm{20}=\mathrm{2}^{\mathrm{2}} ×\mathrm{5} \\ $$$$\mathrm{24}=\mathrm{2}^{\mathrm{3}} ×\mathrm{3} \\ $$$$\Rightarrow\mathrm{2}^{\mathrm{4}} ×\mathrm{3}^{\mathrm{2}} ×\mathrm{5}^{\mathrm{2}} =\mathrm{3600} \\ $$
Commented by zakirullah last updated on 05/Oct/20
$${thanks}\:{alot} \\ $$
Commented by zakirullah last updated on 05/Oct/20
$$\:\:\:{sir}\:{how}\:\mathrm{5}^{\mathrm{2}} \:{comes} \\ $$
Commented by JDamian last updated on 05/Oct/20
it is on purpose. Otherwise the number wouldn't be a perfect squared. In fact, all the primes of the number must be present as an even power.
Answered by $@y@m last updated on 05/Oct/20
$$\mathrm{12}=\:\mathrm{2}×\mathrm{2}×\mathrm{3} \\ $$$$\mathrm{16}=\:\mathrm{2}×\mathrm{2}×\mathrm{2}×\mathrm{2} \\ $$$$\mathrm{20}=\mathrm{2}×\mathrm{2}×\mathrm{5} \\ $$$$\mathrm{24}=\mathrm{2}×\mathrm{2}×\mathrm{2}×\mathrm{3} \\ $$$${All}\:{except}\:\mathrm{5}\:\:{have}\:{pair}. \\ $$$${Therefore}\:{required}\:{no}.{is} \\ $$$$\mathrm{2}×\mathrm{2}×\mathrm{2}×\mathrm{2}×\mathrm{3}×\mathrm{3}×\mathrm{5}×\mathrm{5}=\mathrm{3600} \\ $$
Commented by zakirullah last updated on 05/Oct/20
$${sir}\:{ans}\:{is}\:\mathrm{3600} \\ $$
Commented by zakirullah last updated on 05/Oct/20
$${thanks} \\ $$