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Question Number 163770 by mnjuly1970 last updated on 10/Jan/22

$$$$$$solve:x,y\in \mathbb{N}$$$$$$$$3x+5y=20$$$$-------$$

Answered by MJS_new last updated on 10/Jan/22

$$y=4-\frac{3}{5}x\Rightarrow (x=0\wedge y=4)\vee (x=5\wedge y=1)$$

Commented by MJS_new last updated on 10/Jan/22

$$\mathrm{yes},\mathrm{thank}\mathrm{you}!$$

Commented by MathsFan last updated on 10/Jan/22

$$sir,whichmethoddidyouuseplease$$

Commented by MJS_new last updated on 10/Jan/22

$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{obvious}\mathrm{for}x,y\in \mathbb{N},\mathrm{no}\mathrm{special}\mathrm{method}$$$$\mathrm{needed}.$$$$3x+5y=20\iff y=4-\frac{3}{5}x\Rightarrow$$$$\Rightarrow x=5n\Rightarrow y=4-3n\Rightarrow n=0\vee n=1$$

Answered by Mathspace last updated on 10/Jan/22

$$weconsidercongruencemodulo3$$$$\text{Missing left or extra right}$$$$0+\stackrel{.}{2}\stackrel{.}{y}=\stackrel{.}{2}\Rightarrow \stackrel{.}{y}=\stackrel{.}{1}\Rightarrow y=3k+1$$$$3x=20-5y=20-5(3k+1)$$$$=15-15k(k\in Z)$$$$rememeberthstZ/3Zestuncorps$$

Commented by Mathspace last updated on 10/Jan/22

$$3x=15-15k\Rightarrow x=5-5k$$$$thesetofsolutionis$$$$\left\{(5-5k,1+3k)\right\}$$$$anotherway$$$$ona(5,1)isspecialsolution$$$$because3\times 5+5.1=20wehave$$$$3x+5y=20and3.5+5.1=20\Rightarrow$$$$3(x-1)+5(y-1)=0\Rightarrow$$$$3(x-1)=5(1-y)\Rightarrow$$$$3divide5(1-y)but\mathrm{\Delta }(3,5)=1$$$$bizout\Rightarrow 3/1-y\Rightarrow 1-y=3k\Rightarrow$$$$y=-3k+1$$$$5/3(x-1)\Rightarrow 5/x-1\Rightarrow x=5k+1$$

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