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Question Number 108264 by Rasheed.Sindhi last updated on 15/Aug/20

$$Findoutx,suchthat:$$$$\mathrm{lcm}(50,80,x)=2800\wedge \mathrm{lcm}(56,84,x)=840$$$$$$

Answered by mr W last updated on 15/Aug/20

$$50=2\times 5^2$$$$80=2^4\times 5$$$$2800=2^4\times 5^2\times 7$$$$fromlcm(50,80,x)=2800$$$$\Rightarrow x=2^{0..4}\times 5^{0..2}\times 7^1...\left(i\right)$$$$$$$$56=2^3\times 7$$$$84=2^2\times 3\times 7$$$$840=2^3\times 3\times 5\times 7$$$$fromlcm(56,84,x)=840$$$$\Rightarrow x=2^{0..3}\times 3^{0..1}\times 5^1\times 7^{0..1}...\left(ii\right)$$$$$$$$from\left(i\right)and\left(ii\right):$$$$\Rightarrow x=2^{0..3}\times 3^0\times 5^1\times 7^1=35\times 2^{0..3}$$$$=(35,70,140,280)$$

Commented by Rasheed.Sindhi last updated on 16/Aug/20

$$\mathcal{T}hanksmrWsir!$$

Answered by floor(10²Eta[1]) last updated on 16/Aug/20

$$\mathrm{lcm}(\mathrm{a},\mathrm{b},\mathrm{c})=\mathrm{lcm}(\mathrm{lcm}(\mathrm{a},\mathrm{b}),\mathrm{c})$$$$\left(1\right):\mathrm{lcm}(50,80,\mathrm{x})=\mathrm{lcm}(\mathrm{lcm}(50,80),\mathrm{x})$$$$=\mathrm{lcm}(400,\mathrm{x})=2800$$$$\left(2\right):\mathrm{lcm}(56,84,\mathrm{x})=\mathrm{lcm}(\mathrm{lcm}(56,84),\mathrm{x})$$$$=\mathrm{lcm}(168,\mathrm{x})=840$$$$\left(3\right):400=2^4.5^2$$$$2800=2^4.5^2.7$$$$\Rightarrow \mathrm{x}=2^{\{0,1,2,3,4\}}.5^{\{0,1,2\}}.7$$$$\left(4\right):168=2^3.3.7$$$$840=2^3.3.5.7$$$$\Rightarrow \mathrm{x}=2^{\{0,1,2,3\}}.3^{\{0,1\}}.5.7^{\{0,1\}}$$$$\Rightarrow \mathrm{x}=2^{\{0,1,2,3\}}.5.7$$$$\mathrm{x}=\{35,70,140,280\}$$

Commented by Rasheed.Sindhi last updated on 16/Aug/20

$$\mathcal{T}hanksfloorsir!$$

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