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Question Number 30687 by Rasheed.Sindhi last updated on 24/Feb/18

Given that LCM(A,B,C)=252 LCM(A,B)=36 & LCM(A,C)=63; then: LCM(B,C)=? Pl determine all possible answers.

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{LCM}\left(\mathrm{A},\mathrm{B},\mathrm{C}\right)=\mathrm{252} \\ $$$$\mathrm{LCM}\left(\mathrm{A},\mathrm{B}\right)=\mathrm{36}\:\&\:\mathrm{LCM}\left(\mathrm{A},\mathrm{C}\right)=\mathrm{63}; \\ $$$$\mathrm{then}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{LCM}\left(\mathrm{B},\mathrm{C}\right)=? \\ $$$$\mathrm{Pl}\:\mathrm{determine}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{answers}. \\ $$

Answered by MJS last updated on 24/Feb/18

252=2^2 3^2 7 36=2^2 3^2 63=3^2 7 {A,B}={4,9}∨{9,4}∨{1,36}∨{36,1} {A,C}={7,9}∨{9,7}∨{1,63}∨{63,1} A=9 B=4 C=7 LCM(4,7)=28 or A=1 B=36 C=63 LCM(36,63)=252

$$\mathrm{252}=\mathrm{2}^{\mathrm{2}} \mathrm{3}^{\mathrm{2}} \mathrm{7} \\ $$$$\mathrm{36}=\mathrm{2}^{\mathrm{2}} \mathrm{3}^{\mathrm{2}} \\ $$$$\mathrm{63}=\mathrm{3}^{\mathrm{2}} \mathrm{7} \\ $$$$\left\{{A},{B}\right\}=\left\{\mathrm{4},\mathrm{9}\right\}\vee\left\{\mathrm{9},\mathrm{4}\right\}\vee\left\{\mathrm{1},\mathrm{36}\right\}\vee\left\{\mathrm{36},\mathrm{1}\right\} \\ $$$$\left\{{A},{C}\right\}=\left\{\mathrm{7},\mathrm{9}\right\}\vee\left\{\mathrm{9},\mathrm{7}\right\}\vee\left\{\mathrm{1},\mathrm{63}\right\}\vee\left\{\mathrm{63},\mathrm{1}\right\} \\ $$$${A}=\mathrm{9} \\ $$$${B}=\mathrm{4} \\ $$$${C}=\mathrm{7} \\ $$$${LCM}\left(\mathrm{4},\mathrm{7}\right)=\mathrm{28} \\ $$$$\mathrm{or} \\ $$$$\mathrm{A}=\mathrm{1} \\ $$$${B}=\mathrm{36} \\ $$$${C}=\mathrm{63} \\ $$$${LCM}\left(\mathrm{36},\mathrm{63}\right)=\mathrm{252} \\ $$

Commented by Rasheed.Sindhi last updated on 24/Feb/18

ThαηκS Sir! Are other answers also possible? Could we detefmine all possibilities?

$$\mathcal{T}{h}\alpha\eta\kappa\mathcal{S}\:\mathcal{S}{ir}! \\ $$$$\mathcal{A}\mathrm{re}\:\mathrm{other}\:\mathrm{answers}\:\mathrm{also}\:\mathrm{possible}? \\ $$$$\mathrm{Could}\:\mathrm{we}\:\mathrm{detefmine}\:\mathrm{all}\:\mathrm{possibilities}? \\ $$

Commented by MJS last updated on 24/Feb/18

sorry, even more possibilities: if A or B = 36 the other one can be 1,2,3,4,6,9,12,18 or 36 if A or C = 63 the other one can be 1,3,7,9,21 or 63 so also possible: {A,B,C}={3,36,63}∨{9,36,63} but the LCM(B,C) is 252 too

$$\mathrm{sorry},\:\mathrm{even}\:\mathrm{more}\:\mathrm{possibilities}: \\ $$$$\mathrm{if}\:{A}\:\mathrm{or}\:{B}\:=\:\mathrm{36}\:\mathrm{the}\:\mathrm{other}\:\mathrm{one}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{6},\mathrm{9},\mathrm{12},\mathrm{18}\:\mathrm{or}\:\mathrm{36} \\ $$$$\mathrm{if}\:{A}\:\mathrm{or}\:{C}\:=\:\mathrm{63}\:\mathrm{the}\:\mathrm{other}\:\mathrm{one}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{1},\mathrm{3},\mathrm{7},\mathrm{9},\mathrm{21}\:\mathrm{or}\:\mathrm{63} \\ $$$$\mathrm{so}\:\mathrm{also}\:\mathrm{possible}: \\ $$$$\left\{{A},{B},{C}\right\}=\left\{\mathrm{3},\mathrm{36},\mathrm{63}\right\}\vee\left\{\mathrm{9},\mathrm{36},\mathrm{63}\right\} \\ $$$$\mathrm{but}\:\mathrm{the}\:{LCM}\left({B},{C}\right)\:\mathrm{is}\:\mathrm{252}\:\mathrm{too} \\ $$

Commented by soksan last updated on 25/Feb/18

252=2^2 3^2 7

$$\mathrm{252}=\mathrm{2}^{\mathrm{2}} \mathrm{3}^{\mathrm{2}} \mathrm{7} \\ $$$$ \\ $$

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