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Question Number 72526 by liki last updated on 30/Oct/19

if the lcm and gcf of three numbers are 360 and 6,other numbers are 18 and 60.Find the third number. ... I need help plz...

$${if}\:{the}\:{lcm}\:{and}\:{gcf}\:{of}\:{three}\:{numbers} \\ $$$$\:{are}\:\mathrm{360}\:{and}\:\mathrm{6},{other}\:{numbers}\:{are}\:\mathrm{18}\: \\ $$$$\:{and}\:\mathrm{60}.{Find}\:{the}\:{third}\:{number}. \\ $$$$\:\:\:\:\:\:\:\:...\:{I}\:{need}\:{help}\:{plz}... \\ $$

Commented by TawaTawa last updated on 30/Oct/19

For three numbers a × b × c = ((lcm(a, b, c) × gcd (a, b) × gcd(c, a))/(gcd(a, b, c)))

$$\mathrm{For}\:\mathrm{three}\:\mathrm{numbers} \\ $$$$\:\:\:\:\:\mathrm{a}\:×\:\mathrm{b}\:×\:\mathrm{c}\:\:=\:\:\frac{\mathrm{lcm}\left(\mathrm{a},\:\mathrm{b},\:\mathrm{c}\right)\:×\:\mathrm{gcd}\:\left(\mathrm{a},\:\mathrm{b}\right)\:×\:\mathrm{gcd}\left(\mathrm{c},\:\mathrm{a}\right)}{\mathrm{gcd}\left(\mathrm{a},\:\mathrm{b},\:\mathrm{c}\right)} \\ $$

Answered by mind is power last updated on 30/Oct/19

gcd(a,b,c)=d d∣a,b d∣c d⇒∣gcd(a,b),c⇒d∣gcd((gcd(a,b),c) gcd((gcd(a,b),c)=d⇒d∣c d∣gcd(a,b)⇒d∣a&d∣b ⇒gcd(a,b,c)=gcd(gcd(a,b),c) lcm(a,b,c)=lcm(lcm(a,b),c)) m=lcm(a,b,c)⇒a∣m,b∣m⇒lcm(a,b)∣m c∣m⇒lcm(lcm(a,b),c)∣m d=lcm(lcm(a,b),c)⇒c∣d,lcm(a,b)∣d⇒a∣d,b∣d,c∣d⇒lcm(a,b,c)∣d ⇔lcm(lcm(a,b),c)=lcm(a,b,c) after that a,b,c three number lcm(a,b,c)=360 gcd(a,b,c)=18 a=6,b=18 ⇔ { ((lcm(6,18,c)=360=lcm(lcm(18,60),c)=360)),((gcd(60,18,c)=6=gcd(gcd(60,18),c)=6)) :} lcm(60,18)=6lcm(10,3)=180,gcd(60,18)=6 ⇒lcm(180,c)=360 ⇒gcd(6,c)=6 c=6k⇒gcd(6,6k)=6=6gcd(k,1)=6 lcm(180,6k)=360 6lcm(30,k)=360 lcm(30,k)=60=((30.k)/(gcd(30,k)))⇒2gcd(30,k) =k gcd(30,k)∈{1,2,3,5,6,10,15,30} =1⇒k=2 no gcd=2⇒k=4 ⇒ gcd=3⇒k=6 but gcd(6,30)=6 no gcd=5⇒k=10 gcd(10,30)=10 no gcd=6⇒k=12 gcd=10⇒k=20 gcd=15⇒k=30 non gcd=30⇒k=60 k∈{4,20,60,12} c=6k c∈{360,120,24,72}

$$\mathrm{gcd}\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)=\mathrm{d} \\ $$$$\mathrm{d}\mid\mathrm{a},\mathrm{b} \\ $$$$\mathrm{d}\mid\mathrm{c} \\ $$$$\mathrm{d}\Rightarrow\mid\mathrm{gcd}\left(\mathrm{a},\mathrm{b}\right),\mathrm{c}\Rightarrow\mathrm{d}\mid\mathrm{gcd}\left(\left(\mathrm{gcd}\left(\mathrm{a},\mathrm{b}\right),\mathrm{c}\right)\right. \\ $$$$\mathrm{gcd}\left(\left(\mathrm{gcd}\left(\mathrm{a},\mathrm{b}\right),\mathrm{c}\right)=\mathrm{d}\Rightarrow\mathrm{d}\mid\mathrm{c}\right. \\ $$$$\mathrm{d}\mid\mathrm{gcd}\left(\mathrm{a},\mathrm{b}\right)\Rightarrow\mathrm{d}\mid\mathrm{a\&d}\mid\mathrm{b} \\ $$$$\Rightarrow\mathrm{gcd}\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)=\mathrm{gcd}\left(\mathrm{gcd}\left(\mathrm{a},\mathrm{b}\right),\mathrm{c}\right) \\ $$$$\left.\mathrm{lcm}\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)=\mathrm{lcm}\left(\mathrm{lcm}\left(\mathrm{a},\mathrm{b}\right),\mathrm{c}\right)\right) \\ $$$$\mathrm{m}=\mathrm{lcm}\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)\Rightarrow\mathrm{a}\mid\mathrm{m},\mathrm{b}\mid\mathrm{m}\Rightarrow\mathrm{lcm}\left(\mathrm{a},\mathrm{b}\right)\mid\mathrm{m} \\ $$$$\mathrm{c}\mid\mathrm{m}\Rightarrow\mathrm{lcm}\left(\mathrm{lcm}\left(\mathrm{a},\mathrm{b}\right),\mathrm{c}\right)\mid\mathrm{m} \\ $$$$\mathrm{d}=\mathrm{lcm}\left(\mathrm{lcm}\left(\mathrm{a},\mathrm{b}\right),\mathrm{c}\right)\Rightarrow\mathrm{c}\mid\mathrm{d},\mathrm{lcm}\left(\mathrm{a},\mathrm{b}\right)\mid\mathrm{d}\Rightarrow\mathrm{a}\mid\mathrm{d},\mathrm{b}\mid\mathrm{d},\mathrm{c}\mid\mathrm{d}\Rightarrow\mathrm{lcm}\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)\mid\mathrm{d} \\ $$$$\Leftrightarrow\mathrm{lcm}\left(\mathrm{lcm}\left(\mathrm{a},\mathrm{b}\right),\mathrm{c}\right)=\mathrm{lcm}\left(\mathrm{a},\mathrm{b},\mathrm{c}\right) \\ $$$$\mathrm{after}\:\mathrm{that}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{three}\:\mathrm{number} \\ $$$$\mathrm{lcm}\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)=\mathrm{360} \\ $$$$\mathrm{gcd}\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)=\mathrm{18} \\ $$$$\mathrm{a}=\mathrm{6},\mathrm{b}=\mathrm{18} \\ $$$$\Leftrightarrow\begin{cases}{\mathrm{lcm}\left(\mathrm{6},\mathrm{18},\mathrm{c}\right)=\mathrm{360}=\mathrm{lcm}\left(\mathrm{lcm}\left(\mathrm{18},\mathrm{60}\right),\mathrm{c}\right)=\mathrm{360}}\\{\mathrm{gcd}\left(\mathrm{60},\mathrm{18},\mathrm{c}\right)=\mathrm{6}=\mathrm{gcd}\left(\mathrm{gcd}\left(\mathrm{60},\mathrm{18}\right),\mathrm{c}\right)=\mathrm{6}}\end{cases} \\ $$$$\mathrm{lcm}\left(\mathrm{60},\mathrm{18}\right)=\mathrm{6lcm}\left(\mathrm{10},\mathrm{3}\right)=\mathrm{180},\mathrm{gcd}\left(\mathrm{60},\mathrm{18}\right)=\mathrm{6} \\ $$$$\Rightarrow\mathrm{lcm}\left(\mathrm{180},\mathrm{c}\right)=\mathrm{360} \\ $$$$\Rightarrow\mathrm{gcd}\left(\mathrm{6},\mathrm{c}\right)=\mathrm{6} \\ $$$$\mathrm{c}=\mathrm{6k}\Rightarrow\mathrm{gcd}\left(\mathrm{6},\mathrm{6k}\right)=\mathrm{6}=\mathrm{6gcd}\left(\mathrm{k},\mathrm{1}\right)=\mathrm{6} \\ $$$$\mathrm{lcm}\left(\mathrm{180},\mathrm{6k}\right)=\mathrm{360} \\ $$$$\mathrm{6lcm}\left(\mathrm{30},\mathrm{k}\right)=\mathrm{360} \\ $$$$\mathrm{lcm}\left(\mathrm{30},\mathrm{k}\right)=\mathrm{60}=\frac{\mathrm{30}.\mathrm{k}}{\mathrm{gcd}\left(\mathrm{30},\mathrm{k}\right)}\Rightarrow\mathrm{2gcd}\left(\mathrm{30},\mathrm{k}\right)\:=\mathrm{k} \\ $$$$\mathrm{gcd}\left(\mathrm{30},\mathrm{k}\right)\in\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{6},\mathrm{10},\mathrm{15},\mathrm{30}\right\} \\ $$$$=\mathrm{1}\Rightarrow\mathrm{k}=\mathrm{2}\:\:\:\mathrm{no} \\ $$$$\mathrm{gcd}=\mathrm{2}\Rightarrow\mathrm{k}=\mathrm{4}\:\Rightarrow \\ $$$$\mathrm{gcd}=\mathrm{3}\Rightarrow\mathrm{k}=\mathrm{6}\:\mathrm{but}\:\mathrm{gcd}\left(\mathrm{6},\mathrm{30}\right)=\mathrm{6}\:\:\mathrm{no} \\ $$$$\mathrm{gcd}=\mathrm{5}\Rightarrow\mathrm{k}=\mathrm{10}\:\mathrm{gcd}\left(\mathrm{10},\mathrm{30}\right)=\mathrm{10}\:\mathrm{no} \\ $$$$\mathrm{gcd}=\mathrm{6}\Rightarrow\mathrm{k}=\mathrm{12}\: \\ $$$$\mathrm{gcd}=\mathrm{10}\Rightarrow\mathrm{k}=\mathrm{20}\: \\ $$$$\mathrm{gcd}=\mathrm{15}\Rightarrow\mathrm{k}=\mathrm{30}\:\mathrm{non} \\ $$$$\mathrm{gcd}=\mathrm{30}\Rightarrow\mathrm{k}=\mathrm{60} \\ $$$$\mathrm{k}\in\left\{\mathrm{4},\mathrm{20},\mathrm{60},\mathrm{12}\right\} \\ $$$$\mathrm{c}=\mathrm{6k} \\ $$$$\mathrm{c}\in\left\{\mathrm{360},\mathrm{120},\mathrm{24},\mathrm{72}\right\} \\ $$$$ \\ $$$$ \\ $$

Commented by liki last updated on 30/Oct/19

thanks sir..

$${thanks}\:{sir}.. \\ $$

Commented by mind is power last updated on 30/Oct/19

y′re welcom

$$\mathrm{y}'\mathrm{re}\:\mathrm{welcom} \\ $$

Commented by mr W last updated on 30/Oct/19

24 and 72 also ok

$$\mathrm{24}\:{and}\:\mathrm{72}\:{also}\:{ok} \\ $$

Commented by mind is power last updated on 30/Oct/19

yes i fixed i took 6 but[the two number are 18,60 not 18 and 6 lol

$$\mathrm{yes}\:\mathrm{i}\:\mathrm{fixed}\:\mathrm{i}\:\mathrm{took}\:\mathrm{6}\:\mathrm{but}\left[\mathrm{the}\:\mathrm{two}\:\mathrm{number}\:\mathrm{are}\:\mathrm{18},\mathrm{60}\:\mathrm{not}\:\mathrm{18}\:\mathrm{and}\:\mathrm{6}\right. \\ $$$$\mathrm{lol} \\ $$

Answered by mr W last updated on 30/Oct/19

LCM=360=2^3 ×3^2 ×5^1 GCD=6=2^1 ×3^1 A=18=2^1 ×3^2 B=60=2^2 ×3^1 ×5^1 C=2^3 ×3^(1...2) ×5^(0...1) =8×3=24 =8×3×5=120 =8×9=72 =8×9×5=360

$${LCM}=\mathrm{360}=\mathrm{2}^{\mathrm{3}} ×\mathrm{3}^{\mathrm{2}} ×\mathrm{5}^{\mathrm{1}} \\ $$$${GCD}=\mathrm{6}=\mathrm{2}^{\mathrm{1}} ×\mathrm{3}^{\mathrm{1}} \\ $$$${A}=\mathrm{18}=\mathrm{2}^{\mathrm{1}} ×\mathrm{3}^{\mathrm{2}} \\ $$$${B}=\mathrm{60}=\mathrm{2}^{\mathrm{2}} ×\mathrm{3}^{\mathrm{1}} ×\mathrm{5}^{\mathrm{1}} \\ $$$${C}=\mathrm{2}^{\mathrm{3}} ×\mathrm{3}^{\mathrm{1}...\mathrm{2}} ×\mathrm{5}^{\mathrm{0}...\mathrm{1}} \\ $$$$=\mathrm{8}×\mathrm{3}=\mathrm{24} \\ $$$$=\mathrm{8}×\mathrm{3}×\mathrm{5}=\mathrm{120} \\ $$$$=\mathrm{8}×\mathrm{9}=\mathrm{72} \\ $$$$=\mathrm{8}×\mathrm{9}×\mathrm{5}=\mathrm{360} \\ $$

Commented by TawaTawa last updated on 30/Oct/19

Sir help me check the question i solved in Q72560 if i am correct

$$\mathrm{Sir}\:\mathrm{help}\:\mathrm{me}\:\mathrm{check}\:\mathrm{the}\:\mathrm{question}\:\mathrm{i}\:\mathrm{solved}\:\mathrm{in}\:\mathrm{Q72560} \\ $$$$\mathrm{if}\:\mathrm{i}\:\mathrm{am}\:\mathrm{correct} \\ $$

Commented by liki last updated on 31/Oct/19

•Thank you sir

$$\bullet{Thank}\:{you}\:{sir} \\ $$

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