Question

ex 7 va rog rezolvata si repedeee
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Answer 1

Răspuns:

Explicație pas cu pas:

a)

${x}^{2}(y+2)=192\\D_{192}=\left\{1;2;3;4;6;8;12;16;24;32;48;64;96;192\right\}\\C\breve{a}ut\breve{a}m\:divizorii\:lui\:192\:care\:sunt\:p\breve{a}trate\:perfecte.\\Acestia\:sunt:\:1\:(={1}^{2}),\:4\:(={2}^{2}),16\:(={4}^{2})\:si\:64\:(={8}^{2}).\\Studiem\:cazurile:\\Cazul\:I:x=1\implies{1}^{2}(y+2)=192\implies1(y+2)=192\implies y+2=192\implies y=190\\Cazul\:II:x=2\implies{2}^{2}(y+2)=192\implies4(y+2)=192\implies y+2=192\div4\implies y+2=48\implies y=46$

$Cazul\:III:x=4\implies{4}^{2}(y+2)=192\implies16(y+2)=192\implies y+2=192\div16\implies y+2=12\implies y=10\\Cazul\:IV:x=8\implies{8}^{2}(y+2)=192\implies64(y+2)=192\implies y+2=192\div64\implies y+2=3\implies y=1$

Așadar, (x;y)∈{(1;190) ; (2;46) ; (4;10) ; (8;1)}.

b)

${x}^{2}(2y+1)=180\\D_{180}=\left\{1;2;3;4;5;6;9;10;12;15;18;20;30;36;45;60;90;180\right\}\\C\breve{a}ut\breve{a}m\:divizorii\:lui\:180\:care\:sunt\:p\breve{a}trate\:perfecte.\\Acestia\:sunt:\:1\:(={1}^{2}),\:4\:(={2}^{2}),\:9\:(={3}^{2}),\:36\:(={6}^{2}).\\Cazul\:I:\:x=1\implies{1}^{2}(2y+1)=180\implies1(2y+1)=180\implies2y+1=180\implies2y=179\implies y=\frac{179}{2}\notin\mathbb{N},\:contradictie.$

$Cazul\:II:\:x=2\implies{2}^{2}(2y+1)=180\implies4(2y+1)=180\implies2y+1=180\div4\implies2y+1=45\implies2y=44\implies y=44\div2\implies y=22\\Cazul\:III:\:x=3\implies{3}^{2}(2y+1)=180\implies9(2y+1)=180\implies2y+1=180\div9\implies2y+1=20\implies2y=19\implies y=\frac{19}{2}\notin\mathbb{N},\:contradictie.\\Cazul\:IV:\:x=6\implies{6}^{2}(2y+1)=180\implies36(2y+1)=180\implies2y+1=180\div36\implies2y+1=5\implies2y=4\implies y=4\div2\implies y=2$

Așadar, (x;y)∈{(2;22) ; (6;2)}.