find-atleast-7-solutions-of-the-equation-900x-7689y-109876-CAN-ANYONE-SOLVE-THIS-now-lets-find-7-integral-solutions-

Question Number 84005 by redmiiuser last updated on 08/Mar/20

$${find}\:{atleast}\:\mathrm{7}\:{solutions} \\ $$$${of}\:{the}\:{equation}. \\ $$$$\mathrm{900}{x}+\mathrm{7689}{y}=\mathrm{109876} \\ $$$${CAN}\:{ANYONE}\:{SOLVE} \\ $$$${THIS} \\ $$$${now}\:{lets}\:{find}\:\mathrm{7}\:{integral} \\ $$$${solutions} \\ $$

Commented by redmiiuser last updated on 08/Mar/20

$${i}\:{asked}\:{to}\:{find}\:\mathrm{7}\:{solutions} \\ $$$$ \\ $$

Commented by Rio Michael last updated on 08/Mar/20

or you could use bezouts identity to find the general solution and substitude for some integer λ gen sol: { ((x = x_0 + ((7689)/d)λ)),((y = y_0 −((900)/d)λ)) :} where d = gcd(900,7689)

$$\mathrm{or}\:\mathrm{you}\:\mathrm{could}\:\mathrm{use} \\ $$$$\mathrm{bezouts}\:\mathrm{identity}\:\mathrm{to} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution} \\ $$$$\mathrm{and}\:\mathrm{substitude}\:\mathrm{for}\:\mathrm{some}\:\mathrm{integer}\:\lambda \\ $$$${gen}\:{sol}:\:\:\begin{cases}{{x}\:=\:{x}_{\mathrm{0}} \:+\:\frac{\mathrm{7689}}{{d}}\lambda}\\{{y}\:=\:{y}_{\mathrm{0}} −\frac{\mathrm{900}}{{d}}\lambda}\end{cases} \\ $$$${where}\:{d}\:=\:{gcd}\left(\mathrm{900},\mathrm{7689}\right) \\ $$

Commented by mr W last updated on 08/Mar/20

there is no integral solution! since gcd(900,7689)=3≠1, that means LHS is always a multiple of 3, but 109876 is not a multiple of 3.

$${there}\:{is}\:{no}\:{integral}\:{solution}!\:{since} \\ $$$${gcd}\left(\mathrm{900},\mathrm{7689}\right)=\mathrm{3}\neq\mathrm{1},\:{that}\:{means} \\ $$$${LHS}\:{is}\:{always}\:{a}\:{multiple}\:{of}\:\mathrm{3},\:{but} \\ $$$$\mathrm{109876}\:{is}\:{not}\:{a}\:{multiple}\:{of}\:\mathrm{3}. \\ $$

Commented by Rio Michael last updated on 08/Mar/20

$$\mathrm{thats}\:\mathrm{a}\:\mathrm{good}\:\mathrm{point}\:\mathrm{sir}\:\mathrm{thanks} \\ $$

Answered by $@ty@m123 last updated on 08/Mar/20

As there is no restrictions, we can put x=1,2,3, .....up to ∞ and find corresponding values of y.

$${As}\:{there}\:{is}\:{no}\:{restrictions}, \\ $$$${we}\:{can}\:{put}\:{x}=\mathrm{1},\mathrm{2},\mathrm{3},\:…..{up}\:{to}\:\infty \\ $$$${and}\:{find}\:{corresponding}\:{values}\:{of}\:{y}. \\ $$

Commented by redmiiuser last updated on 08/Mar/20