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Question Number 41003 by mondodotto@gmail.com last updated on 30/Jul/18

The LCM and GCF of x,18 and 60 are 360 and 6 respectively. find the value of x

$$\boldsymbol{\mathrm{The}}\:\boldsymbol{\mathrm{LCM}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{GCF}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{{x}},\mathrm{18}\:\boldsymbol{\mathrm{and}}\:\mathrm{60} \\ $$$$\boldsymbol{\mathrm{are}}\:\mathrm{360}\:\boldsymbol{\mathrm{and}}\:\mathrm{6}\:\boldsymbol{\mathrm{respectively}}. \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{{x}} \\ $$

Answered by $@ty@m last updated on 31/Jul/18

Let x=6k We have 18=6×3 60=6×10 ∵LCM=360 ∴6×k×3×10=360 ⇒k=2 ⇒x=12

$${Let}\:{x}=\mathrm{6}{k} \\ $$$${We}\:{have}\:\mathrm{18}=\mathrm{6}×\mathrm{3} \\ $$$$\mathrm{60}=\mathrm{6}×\mathrm{10} \\ $$$$\because{LCM}=\mathrm{360} \\ $$$$\therefore\mathrm{6}×{k}×\mathrm{3}×\mathrm{10}=\mathrm{360} \\ $$$$\Rightarrow{k}=\mathrm{2} \\ $$$$\Rightarrow{x}=\mathrm{12} \\ $$

Commented by MrW3 last updated on 31/Jul/18

LCM from 12, 18 and 60 is 180, not 360, so the answer is not correct.

$${LCM}\:{from}\:\mathrm{12},\:\mathrm{18}\:{and}\:\mathrm{60}\:{is}\:\mathrm{180},\:{not} \\ $$$$\mathrm{360},\:{so}\:{the}\:{answer}\:{is}\:{not}\:{correct}. \\ $$

Commented by mondodotto@gmail.com last updated on 31/Jul/18

possible answer not oly 12 there is some other

$$\mathrm{possible}\:\mathrm{answer}\:\mathrm{not}\:\mathrm{oly}\:\mathrm{12}\:\mathrm{there}\:\mathrm{is} \\ $$$$\mathrm{some}\:\mathrm{other}\: \\ $$

Answered by MJS last updated on 01/Aug/18

18=2×3^2 60=2^2 ×3×5 360=2^3 ×3^2 ×5 ⇒ x=2^3 ×3^(0≤j≤2) ×5^(0≤k≤1) 6=2×3 ⇒ x=2^(i≥1) ×3^(j≥1) ×5^(k≥0) i=3 j∈{1, 2} k∈{0, 1} x∈{3, 3^2 , 3×5, 3^2 ×5}×2^3 = ={24, 72, 120, 360}

$$\mathrm{18}=\mathrm{2}×\mathrm{3}^{\mathrm{2}} \\ $$$$\mathrm{60}=\mathrm{2}^{\mathrm{2}} ×\mathrm{3}×\mathrm{5} \\ $$$$\mathrm{360}=\mathrm{2}^{\mathrm{3}} ×\mathrm{3}^{\mathrm{2}} ×\mathrm{5} \\ $$$$\Rightarrow\:{x}=\mathrm{2}^{\mathrm{3}} ×\mathrm{3}^{\mathrm{0}\leqslant{j}\leqslant\mathrm{2}} ×\mathrm{5}^{\mathrm{0}\leqslant{k}\leqslant\mathrm{1}} \\ $$$$\mathrm{6}=\mathrm{2}×\mathrm{3} \\ $$$$\Rightarrow\:{x}=\mathrm{2}^{{i}\geqslant\mathrm{1}} ×\mathrm{3}^{{j}\geqslant\mathrm{1}} ×\mathrm{5}^{{k}\geqslant\mathrm{0}} \\ $$$${i}=\mathrm{3} \\ $$$${j}\in\left\{\mathrm{1},\:\mathrm{2}\right\} \\ $$$${k}\in\left\{\mathrm{0},\:\mathrm{1}\right\} \\ $$$${x}\in\left\{\mathrm{3},\:\mathrm{3}^{\mathrm{2}} ,\:\mathrm{3}×\mathrm{5},\:\mathrm{3}^{\mathrm{2}} ×\mathrm{5}\right\}×\mathrm{2}^{\mathrm{3}} = \\ $$$$=\left\{\mathrm{24},\:\mathrm{72},\:\mathrm{120},\:\mathrm{360}\right\} \\ $$

Commented by mondodotto@gmail.com last updated on 01/Aug/18

thank you sir,blessed

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir},\mathrm{blessed} \\ $$$$ \\ $$

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