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

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

Answered by MJS last updated on 24/Feb/18

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

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

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

Commented by MJS last updated on 24/Feb/18

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

Commented by soksan last updated on 25/Feb/18

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

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