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Question Number 136830 by bramlexs22 last updated on 26/Mar/21

$$$$ Determine all solutions in the integers of the following Diophantine equations: (a)56x+72y=40 (b)24x+138y=18 (c)221x+35y=11\n

Answered by floor(10²Eta[1]) last updated on 26/Mar/21

$$\text{Extra close brace or missing open brace}$$ $$\Rightarrow 7\mathrm{x}+9\mathrm{y}=5$$ $$7\mathrm{x}+9\mathrm{y}=5\left(\mathrm{look}\mathrm{mod}3\right)$$ $$\mathrm{x}\equiv 2\left(\mathrm{mod}3\right)\Rightarrow \mathrm{x}=3\mathrm{a}+2$$ $$7(3\mathrm{k}+2)+9\mathrm{y}=5$$ $$\mathrm{y}=-1-\frac{7}{3}\mathrm{a}\Rightarrow 3\mid \mathrm{a}\Rightarrow \mathrm{a}=3\mathrm{k}$$ $$\mathrm{x}=9\mathrm{k}+2,\mathrm{y}=-1-7\mathrm{k}$$ $$\text{Extra close brace or missing open brace}$$ $$\Rightarrow 4\mathrm{x}+23\mathrm{y}=3\left(\mathrm{look}\mathrm{mod}2\right)$$ $$\mathrm{y}\equiv 1\left(\mathrm{mod}2\right)\Rightarrow \mathrm{y}=2\mathrm{a}+1$$ $$4\mathrm{x}+23(2\mathrm{a}+1)=3$$ $$\mathrm{x}=-5-\frac{23}{2}\mathrm{a}\Rightarrow 2\mid \mathrm{a}\Rightarrow \mathrm{a}=2\mathrm{k}$$ $$\mathrm{x}=-5-23\mathrm{k},\mathrm{y}=4\mathrm{k}+1$$ $$\text{Extra close brace or missing open brace}$$ $$\mathrm{x}\equiv 1\left(\mathrm{mod}5\right)\Rightarrow \mathrm{x}=5\mathrm{a}+1$$ $$221(5\mathrm{a}+1)+35\mathrm{y}=11$$ $$\mathrm{y}=-\frac{221}{7}\mathrm{a}-6\Rightarrow 7\mid \mathrm{a}\Rightarrow \mathrm{a}=7\mathrm{k}$$ $$\mathrm{x}=35\mathrm{k}+1,\mathrm{y}=-221\mathrm{k}-6$$ $$$$

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