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Question Number 146488 by mathdanisur last updated on 13/Jul/21

if x+y=216 and dcm(x;y)=18 find x−y=?

$${if}\:\:\:{x}+{y}=\mathrm{216}\:\:\:{and}\:\:\:\boldsymbol{{dcm}}\left({x};{y}\right)=\mathrm{18} \\ $$$${find}\:\:\:{x}−{y}=? \\ $$

Commented by mathdanisur last updated on 13/Jul/21

the biggest common divisot

$${the}\:{biggest}\:{common}\:{divisot} \\ $$

Commented by gsk2684 last updated on 13/Jul/21

dcm means?

$${dcm}\:{means}? \\ $$

Answered by mathmax by abdo last updated on 13/Jul/21

x=qd and y=q^′ d with Δ(q,q^′ )=1 (d=18) x+y=216 ⇒qd +q^′ d=216 ⇒q+q^′ =((216)/(18))=13 we get the system { ((q+q^′ =13)),((Δ(q,q^′ )=1)) :} q=1 ⇒q^′ =12 sol^o ⇒x=18 and y=12×18=... q=2⇒q^′ =11 sol^o ⇒x=36 and y=11×18=... q=3⇒q^′ =8 sol^o ⇒x=3×18 and y=8×18 q=4 ⇒q^′ =9 sol^o ⇒x=4×18 and y=9×18 q=5 ⇒q^′ =8 sol^o ⇒x=5×18 and y=8×18 q=6 ⇒q^′ =7 sol^o ⇒x=6×18 and y=7×18 now you can find x−y....(dont forget symetrie of the system!)

$$\mathrm{x}=\mathrm{qd}\:\mathrm{and}\:\mathrm{y}=\mathrm{q}^{'} \mathrm{d}\:\mathrm{with}\:\Delta\left(\mathrm{q},\mathrm{q}^{'} \right)=\mathrm{1}\:\:\:\:\left(\mathrm{d}=\mathrm{18}\right) \\ $$$$\mathrm{x}+\mathrm{y}=\mathrm{216}\:\Rightarrow\mathrm{qd}\:+\mathrm{q}^{'} \:\mathrm{d}=\mathrm{216}\:\Rightarrow\mathrm{q}+\mathrm{q}^{'} \:=\frac{\mathrm{216}}{\mathrm{18}}=\mathrm{13} \\ $$$$\mathrm{we}\:\mathrm{get}\:\mathrm{the}\:\mathrm{system}\:\:\begin{cases}{\mathrm{q}+\mathrm{q}^{'} \:=\mathrm{13}}\\{\Delta\left(\mathrm{q},\mathrm{q}^{'} \right)=\mathrm{1}}\end{cases}\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\mathrm{q}=\mathrm{1}\:\Rightarrow\mathrm{q}^{'} =\mathrm{12}\:\:\mathrm{sol}^{\mathrm{o}} \:\Rightarrow\mathrm{x}=\mathrm{18}\:\mathrm{and}\:\mathrm{y}=\mathrm{12}×\mathrm{18}=... \\ $$$$\mathrm{q}=\mathrm{2}\Rightarrow\mathrm{q}^{'} \:=\mathrm{11}\:\mathrm{sol}^{\mathrm{o}} \:\Rightarrow\mathrm{x}=\mathrm{36}\:\mathrm{and}\:\mathrm{y}=\mathrm{11}×\mathrm{18}=... \\ $$$$\mathrm{q}=\mathrm{3}\Rightarrow\mathrm{q}^{'} \:=\mathrm{8}\:\mathrm{sol}^{\mathrm{o}} \:\Rightarrow\mathrm{x}=\mathrm{3}×\mathrm{18}\:\:\mathrm{and}\:\mathrm{y}=\mathrm{8}×\mathrm{18} \\ $$$$\mathrm{q}=\mathrm{4}\:\Rightarrow\mathrm{q}^{'} \:=\mathrm{9}\:\:\mathrm{sol}^{\mathrm{o}} \:\Rightarrow\mathrm{x}=\mathrm{4}×\mathrm{18}\:\mathrm{and}\:\mathrm{y}=\mathrm{9}×\mathrm{18} \\ $$$$\mathrm{q}=\mathrm{5}\:\Rightarrow\mathrm{q}^{'} \:=\mathrm{8}\:\mathrm{sol}^{\mathrm{o}} \:\Rightarrow\mathrm{x}=\mathrm{5}×\mathrm{18}\:\mathrm{and}\:\mathrm{y}=\mathrm{8}×\mathrm{18} \\ $$$$\mathrm{q}=\mathrm{6}\:\Rightarrow\mathrm{q}^{'} \:=\mathrm{7}\:\:\mathrm{sol}^{\mathrm{o}} \Rightarrow\mathrm{x}=\mathrm{6}×\mathrm{18}\:\mathrm{and}\:\mathrm{y}=\mathrm{7}×\mathrm{18} \\ $$$$\mathrm{now}\:\mathrm{you}\:\mathrm{can}\:\mathrm{find}\:\mathrm{x}−\mathrm{y}....\left(\mathrm{dont}\:\mathrm{forget}\:\mathrm{symetrie}\:\mathrm{of}\:\mathrm{the}\:\mathrm{system}!\right) \\ $$$$ \\ $$$$\:\: \\ $$

Commented by mathdanisur last updated on 13/Jul/21

thanks Ser, but answer 90;126

$${thanks}\:{Ser},\:{but}\:{answer}\:\mathrm{90};\mathrm{126} \\ $$

Commented by Rasheed.Sindhi last updated on 15/Jul/21

mathmax sir In 2nd line a typo: x+y=216 ⇒qd +q^′ d=216 ⇒q+q^′ =((216)/(18))=12

$${mathmax}\:{sir} \\ $$$${In}\:\mathrm{2}{nd}\:{line}\:{a}\:{typo}: \\ $$$$\mathrm{x}+\mathrm{y}=\mathrm{216}\:\Rightarrow\mathrm{qd}\:+\mathrm{q}^{'} \:\mathrm{d}=\mathrm{216}\:\Rightarrow\mathrm{q}+\mathrm{q}^{'} \:=\frac{\mathrm{216}}{\mathrm{18}}=\mathrm{12} \\ $$

Answered by Rasheed.Sindhi last updated on 13/Jul/21

x+y=216 and gcd(x,y)=18 Let x=18u & y=18v where gcd(u,v)=1 x+y=216⇒18u+18v=216 ⇒u+v=12 (u,v)=(1,11)=(5,7)=(7,5)=(11,1) u−v=−10,−2,2,10 x−y=18u−18v=18(u−v) =18(−10),18(−2),18(2),18(10) x−y=−180,−36,36,180

$$\:\:\:{x}+{y}=\mathrm{216}\:\:\:{and}\:\:\:\boldsymbol{{gcd}}\left({x},{y}\right)=\mathrm{18} \\ $$$${Let}\:{x}=\mathrm{18}{u}\:\&\:{y}=\mathrm{18}{v}\:{where}\:\boldsymbol{{gcd}}\left({u},{v}\right)=\mathrm{1} \\ $$$${x}+{y}=\mathrm{216}\Rightarrow\mathrm{18}{u}+\mathrm{18}{v}=\mathrm{216} \\ $$$$\Rightarrow{u}+{v}=\mathrm{12} \\ $$$$\:\:\:\:\left({u},{v}\right)=\left(\mathrm{1},\mathrm{11}\right)=\left(\mathrm{5},\mathrm{7}\right)=\left(\mathrm{7},\mathrm{5}\right)=\left(\mathrm{11},\mathrm{1}\right) \\ $$$${u}−{v}=−\mathrm{10},−\mathrm{2},\mathrm{2},\mathrm{10} \\ $$$${x}−{y}=\mathrm{18}{u}−\mathrm{18}{v}=\mathrm{18}\left({u}−{v}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\mathrm{18}\left(−\mathrm{10}\right),\mathrm{18}\left(−\mathrm{2}\right),\mathrm{18}\left(\mathrm{2}\right),\mathrm{18}\left(\mathrm{10}\right) \\ $$$${x}−{y}=−\mathrm{180},−\mathrm{36},\mathrm{36},\mathrm{180} \\ $$

Commented by mathdanisur last updated on 13/Jul/21

thanks Ser, but 90;126

$${thanks}\:{Ser},\:{but}\:\mathrm{90};\mathrm{126} \\ $$

Commented by Rasheed.Sindhi last updated on 13/Jul/21

Your answers seem wrong to me. x+y=216(given) If x−y=90(your answer) x=153 ,y=63 gcd(153,63)=9≠18 Similarly x+y=216(given) If x−y=126(your answer) x=171 ,y=45 gcd(171,45)=9≠18

$${Your}\:{answers}\:{seem}\:{wrong}\:{to}\:{me}. \\ $$$$\:{x}+{y}=\mathrm{216}\left({given}\right) \\ $$$${If}\:{x}−{y}=\mathrm{90}\left({your}\:{answer}\right) \\ $$$$\:\:{x}=\mathrm{153}\:,{y}=\mathrm{63} \\ $$$$\boldsymbol{{gcd}}\left(\mathrm{153},\mathrm{63}\right)=\mathrm{9}\neq\mathrm{18} \\ $$$$\:{Similarly} \\ $$$$\:\:\:\:{x}+{y}=\mathrm{216}\left({given}\right) \\ $$$${If}\:{x}−{y}=\mathrm{126}\left({your}\:{answer}\right) \\ $$$${x}=\mathrm{171}\:,{y}=\mathrm{45} \\ $$$$\:\:\:\boldsymbol{{gcd}}\left(\mathrm{171},\mathrm{45}\right)=\mathrm{9}\neq\mathrm{18} \\ $$

Commented by mathdanisur last updated on 13/Jul/21

good Ser, thanks

$${good}\:{Ser},\:{thanks} \\ $$

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