Question

Determinaţi numerele naturale a şi b pentru care 17a + 19 = 518.
va rog frumos ajutor la rezolvarea acestei probleme! ​

Answer 1

$\displaystyle\bf\\17a+19b=518,~a,b\in\mathbb{N}.\\plecam~de~la~faptul~ca~518=\mathcal{M}_{19}+5.\\deci,~17a+19b=\mathcal{M}_{19}+5 \Leftrightarrow 17a=\mathcal{M}_{19}+5,~dar~17=\mathcal{M}_{19}+17,~de~\\unde~ne~rezulta~ca~a=\mathcal{M}_{19}+7,~notam~a=19k+7,~k~\in\mathbb{N}.\\17a+19b=17(19k+7)+19b=323k+119+19b=518~|-119 \implies \\323k+19b=399~|:19 \implies 17k+b=21 \Leftrightarrow 17k=21-b,~de~unde~\\21-b~poate~fi~0~sau~17~\implies b~poate~fi~21~sau~4.\\daca~b=21\implies a=7.\\daca~b=4 \implies a=26.$