Răspuns: [tex]\purple{\boxed{\bf \overline{abc}\in \big\{105;113;121;204;212;220;303;311;402;410;501;600\big\}}}[/tex]
Explicație pas cu pas:
[tex]\bf ~[/tex]
[tex]\bf ~\overline{abc}~numerele ~cautate[/tex]
a, b, c = cifre
[tex]\bf a,b,c,~\in \{ 0,1,2,3,4,5,6,7,8,9\}[/tex]
[tex]\bf a\neq 0[/tex]
[tex]\bf ~[/tex]
[tex]\bf Caz~ I)~\green{\underline{a = 1}} \Rightarrow 1 +2b+c=6\Rightarrow 2b+c=6-1\Rightarrow \underline{2b+c= 5}[/tex]
[tex]\bf Daca~\underline{b = 0} \Rightarrow2\cdot0+c= 5 \Rightarrow\underline{c = 5} \Rightarrow \red{\boxed{\bf \overline{abc}=105}}[/tex]
[tex]\bf Daca~\underline{b = 1} \Rightarrow2\cdot1+c= 5 \Rightarrow\underline{c = 3} \Rightarrow \red{\boxed{\bf \overline{abc}=113}}[/tex]
[tex]\bf Daca~\underline{b = 2} \Rightarrow2\cdot2+c= 5 \Rightarrow\underline{c = 1} \Rightarrow \red{\boxed{\bf \overline{abc}=121}}[/tex]
[tex]\bf ~[/tex]
[tex]\bf Caz~ II)~ \green{\underline{a = 2}} \Rightarrow 2 +2b+c=6\Rightarrow \underline{2b+c= 4}[/tex]
[tex]\bf Daca~\underline{b = 0} \Rightarrow2\cdot0+c= 4 \Rightarrow\underline{c = 4} \Rightarrow \red{\boxed{\bf \overline{abc}=204}}[/tex]
[tex]\bf Daca~\underline{b = 1} \Rightarrow2\cdot1+c= 4 \Rightarrow\underline{c = 2} \Rightarrow \red{\boxed{\bf \overline{abc}=212}}[/tex]
[tex]\bf Daca~\underline{b = 2} \Rightarrow2\cdot2+c= 4 \Rightarrow\underline{c = 0} \Rightarrow \red{\boxed{\bf \overline{abc}=220}}[/tex]
[tex]\bf ~[/tex]
[tex]\bf Caz~ III)~\green{\underline{a =3}} \Rightarrow 3 +2b+c=6\Rightarrow 2b+c=6-3\Rightarrow \underline{2b+c= 3}[/tex]
[tex]\bf Daca~\underline{b = 0} \Rightarrow2\cdot0+c= 3 \Rightarrow\underline{c =3} \Rightarrow \red{\boxed{\bf \overline{abc}=303}}[/tex]
[tex]\bf Daca~\underline{b = 1} \Rightarrow2\cdot1+c= 3 \Rightarrow\underline{c = 1} \Rightarrow \red{\boxed{\bf \overline{abc}=311}}[/tex]
[tex]\bf ~[/tex]
[tex]\bf Caz~ IV)~ \green{\underline{a =4}} \Rightarrow 4 +2b+c=6\Rightarrow 2b+c=6-4\Rightarrow \underline{2b+c= 2}[/tex]
[tex]\bf Daca~\underline{b = 0} \Rightarrow2\cdot0+c= 2 \Rightarrow\underline{c =2} \Rightarrow \red{\boxed{\bf \overline{abc}=402}}[/tex]
[tex]\bf Daca~\underline{b = 1} \Rightarrow2\cdot1+c= 2 \Rightarrow\underline{c = 0} \Rightarrow \red{\boxed{\bf \overline{abc}=410}}[/tex]
[tex]\bf~[/tex]
[tex]\bf Caz~ V)~ \green{\underline{a =5}} \Rightarrow 5 +2b+c=6\Rightarrow \underline{2b+c= 1}[/tex]
[tex]\bf Daca~\underline{b = 0} \Rightarrow2\cdot0+c= 1 \Rightarrow\underline{c =1} \Rightarrow \red{\boxed{\bf \overline{abc}=501}}[/tex]
[tex]\bf~[/tex]
[tex]\bf Caz~ VI)~ \green{\underline{a =6}} \Rightarrow 6 +2b+c=6\Rightarrow \underline{2b+c=6}[/tex]
[tex]\bf Daca~\underline{b = 0} \Rightarrow2\cdot0+c= 0 \Rightarrow\underline{c =0} \Rightarrow \red{\boxed{\bf \overline{abc}=600}}[/tex]
[tex]\bf ~[/tex]
Numere naturale sunt:[tex]\purple{\boxed{\bf \overline{abc}\in \big\{105;113;121;204;212;220;303;311;402;410;501;600\big\}}}[/tex]
==pav38==
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