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Question Number 188970 by BaliramKumar last updated on 09/Mar/23

LCM(x, 144, 150) = 10800 how many value of x.

$$\mathrm{LCM}\left({x},\:\mathrm{144},\:\mathrm{150}\right)\:=\:\mathrm{10800}\:{how}\:{many}\:{value}\:{of}\:\:{x}. \\ $$$$ \\ $$

Answered by mr W last updated on 10/Mar/23

10800=2^4 ×3^3 ×5^2 144=2^4 ×3^2 150=2×3×5^2 x=2^(0−4) ×3^3 ×5^(0−2) ⇒there are 5×3=15 possible values for x.

$$\mathrm{10800}=\mathrm{2}^{\mathrm{4}} ×\mathrm{3}^{\mathrm{3}} ×\mathrm{5}^{\mathrm{2}} \\ $$$$\mathrm{144}=\mathrm{2}^{\mathrm{4}} ×\mathrm{3}^{\mathrm{2}} \\ $$$$\mathrm{150}=\mathrm{2}×\mathrm{3}×\mathrm{5}^{\mathrm{2}} \\ $$$${x}=\mathrm{2}^{\mathrm{0}−\mathrm{4}} ×\mathrm{3}^{\mathrm{3}} ×\mathrm{5}^{\mathrm{0}−\mathrm{2}} \\ $$$$\Rightarrow{there}\:{are}\:\mathrm{5}×\mathrm{3}=\mathrm{15}\:{possible}\:{values} \\ $$$${for}\:{x}. \\ $$

Commented by BaliramKumar last updated on 10/Mar/23

Thanks

$${Thanks} \\ $$

Commented by BaliramKumar last updated on 10/Mar/23

please explain last step

$${please}\:{explain}\:\:{last}\:{step} \\ $$

Commented by mr W last updated on 10/Mar/23

x=a×3^3 ×b a=(2^0 ,2^1 ,2^2 ,2^3 ,2^4 ) b=(5^0 ,5^1 ,5^2 ) a has 5 possibilities b has 3 possibilities so x=a×3^3 ×b has 5×3=15 possibilities.

$${x}={a}×\mathrm{3}^{\mathrm{3}} ×{b} \\ $$$${a}=\left(\mathrm{2}^{\mathrm{0}} ,\mathrm{2}^{\mathrm{1}} ,\mathrm{2}^{\mathrm{2}} ,\mathrm{2}^{\mathrm{3}} ,\mathrm{2}^{\mathrm{4}} \right) \\ $$$${b}=\left(\mathrm{5}^{\mathrm{0}} ,\mathrm{5}^{\mathrm{1}} ,\mathrm{5}^{\mathrm{2}} \right) \\ $$$${a}\:{has}\:\mathrm{5}\:{possibilities} \\ $$$${b}\:{has}\:\mathrm{3}\:{possibilities} \\ $$$${so}\:{x}={a}×\mathrm{3}^{\mathrm{3}} ×{b}\:{has}\:\mathrm{5}×\mathrm{3}=\mathrm{15}\:{possibilities}. \\ $$

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