Pagina:Gauss, Carl Friedrich - Werke (1870).djvu/38

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de congruentiis primi gradus

${\begin{alignedat}{1}a\zeta &+a'\zeta '&+a''\zeta ''&+\mathrm {etc.} =0\\b\zeta &+b'\zeta '&+b''\zeta ''&+\mathrm {etc.} =0\\&&\mathrm {etc.} &\mathrm {etc.} \end{alignedat}}$

3) Manifestum est, si congruentiae $A,A',A''\!$ etc. per $\xi ,\xi ',\xi ''\!$ etc., tum per $\upsilon ,\upsilon ',\upsilon ''\!$ etc. etc. multiplicentur, tuncque addantur, has congruentias proventuras esse:$\textstyle {\begin{alignedat}{1}(a\xi &+a'\xi '&+a''\xi ''&+\mathrm {etc.} )\ x\ &\equiv f\xi &+f'\xi '&+f''\xi ''&+\mathrm {etc.} \\(b\upsilon &+b'\upsilon '&+b''\upsilon ''&+\mathrm {etc.} )\ y\ &\equiv f\upsilon &+f'\upsilon '&+f''\upsilon ''&+\mathrm {etc.} \\(c\zeta &+c'\zeta '&+c''\zeta ''&+\mathrm {etc.} )\ z\ &\equiv f\zeta &+f'\zeta '&+f''\zeta ''&+\mathrm {etc.} \\&&&&\mathrm {etc.} \end{alignedat}}$quas brevitatis gratia ita exhibemus:

$$\textstyle \Sigma (a\xi )x\equiv \Sigma (f\xi ),\quad \Sigma (b\upsilon )y\equiv \Sigma (f\upsilon ),\quad \Sigma (c\zeta )z\equiv \Sigma (f\zeta ),\quad \mathrm {etc.}$$

4) Iam plures casus sunt distinguendi.

Primo quando omnes incognitarum coefficientes $\Sigma (a\xi )$, $\Sigma (b\upsilon )$ etc. ad congruentiarum modulum $m$ sunt primi, hae congruentiae secundum praecepta ante tradita solvi possunt, problematisque solutio completa per congruentias formae $\textstyle x\equiv p{\pmod {m}}$, $\textstyle y\equiv q{\pmod {m}}$ etc. exhibebitur^[1]. E. g. Si proponuntur congruentiae $\textstyle x+3y+z\equiv 1,\quad 4x+y+5z\equiv 7,\quad 2x+2y+z\equiv 3{\pmod {8}}$invenietur $\xi =9$, $\xi '=1$, $\xi ''=-14$, unde fit $-15x\equiv 26$, quare $\textstyle x\equiv 6{\pmod {8}}$; eodem modo invenitur $15y\equiv -4$, $15z\equiv 1$, et hinc $y\equiv 4$, $\textstyle z\equiv 7{\pmod {8}}$.

5) Secundo quando non omnes coefficientes $\Sigma (a\xi )$, $\Sigma (b\upsilon )$ etc. ad modulum sunt primi , sint $\alpha$, $\beta$, $\gamma$ etc. divisores communes maximi ipsius $m$ cum $\Sigma (a\xi )$, $\Sigma (b\upsilon )$, $\Sigma (c\zeta )$ etc. resp., patetque problema impossibile esse, nisi illi numeros $\Sigma (f\xi )$, $\Sigma (f\upsilon )$, $\Sigma (f\zeta )$ etc. resp. metiantur. Quando vero hae conditiones locum habent, congruentiae in (3) complete resolventur per tales $\textstyle x\equiv p{\pmod {\frac {m}{\alpha }}}$, $\textstyle y\equiv q{\pmod {\frac {m}{\beta }}}$, $\textstyle z\equiv r{\pmod {\frac {m}{\gamma }}}$ etc., aut si mavis dabuntur $\alpha$ valores diversi ipsius $x$ (i. e. secundum $m$ incongrui, puta $p$, $\textstyle p+{\frac {m}{\alpha }}$ … $\textstyle p+{\frac {(\alpha -1)m}{\alpha }}$),

1. Observare convenit hancce conclusionem demonstratione egere, quam autem hic supprimimus. Proprie enim nihil aliud ex analysi nostra sequitur, quam quod congruentiae propositae per alios incognitarum $x$, $y$ etc. valores solvi nequeant: hos vero satisfacere non sequitur. Fieri enim posset, ut nulla omnino solutio daretur. Similis paralogismus etiam in aequationum linearium explicatione plerumque committitur.