Răspuns: [tex]\pink{\boxed{\bf \overline{ab}\in \big\{18;27;36;45;54;63;72;81\big\}}}[/tex]
Explicație pas cu pas:
[tex]\bf ~[/tex]
[tex]\bf ~\overline{ab}~numerele ~cautate[/tex]
a, b = cifre
[tex]\bf a,b\in \{ 0,1,2,3,4,5,6,7,8,9\}[/tex]
[tex]\bf a\neq 0~~~b\neq 0[/tex]
[tex]\bf \overline{ab}+\overline{ba}= 99[/tex]
Descompunem în bază zece și vom avea:
[tex]\bf 10a+b+10b+a=99[/tex]
[tex]\bf 11a+11b=99~~~~\bigg|:11[/tex]
[tex]\bf \underline{a+b=9}[/tex]
Analizăm în funcție de ce valoare poate lua a
[tex]\bf ~ I)~\blue{\underline{a = 1}} \Rightarrow 1 +b=9\Rightarrow b=9-1\Rightarrow \blue{\underline{b=8}}[/tex]
[tex] \Rightarrow \red{\boxed{\bf \overline{ab}=18}}[/tex]
[tex] \Rightarrow \red{\bf \overline{ba}=81}[/tex]
[tex]\bf ~[/tex]
[tex]\bf ~ II)~\blue{\underline{a = 2}} \Rightarrow 2 +b=9\Rightarrow b=9-2\Rightarrow \blue{\underline{b=7}}[/tex]
[tex] \Rightarrow \red{\boxed{\bf \overline{ab}=27}}[/tex]
[tex] \Rightarrow \red{\bf \overline{ba}=72}[/tex]
[tex]\bf ~[/tex]
[tex]\bf ~ III)~\blue{\underline{a = 3}} \Rightarrow 3 +b=9\Rightarrow b=9-3\Rightarrow \blue{\underline{b=6}}[/tex]
[tex] \Rightarrow \red{\boxed{\bf \overline{ab}=36}}[/tex]
[tex] \Rightarrow \red{\bf \overline{ba}=63}[/tex]
[tex]\bf ~[/tex]
[tex]\bf ~ IV)~\blue{\underline{a = 4}} \Rightarrow 4 +b=9\Rightarrow b=9-4\Rightarrow \blue{\underline{b=5}}[/tex]
[tex] \Rightarrow \red{\boxed{\bf \overline{ab}=45}[/tex]
[tex] \Rightarrow \red{\bf \overline{ba}=54}[/tex]
[tex]\bf ~[/tex]
[tex]\bf ~ V)~\blue{\underline{a = 5}} \Rightarrow 5 +b=9\Rightarrow b=9-5\Rightarrow \underline{b=4}[/tex]
[tex] \Rightarrow \red{\boxed{\bf \overline{ab}=54}}[/tex]
[tex] \Rightarrow \red{\bf \overline{ba}=45}[/tex]
[tex]\bf ~[/tex]
[tex]\bf ~ VI)~\blue{\underline{a = 6}} \Rightarrow 6 +b=9\Rightarrow b=9-6\Rightarrow \blue{\underline{b=3}}[/tex]
[tex] \Rightarrow \red{\boxed{\bf \overline{ab}=63}}[/tex]
[tex] \Rightarrow \red{\bf \overline{ba}=36}[/tex]
[tex]\bf ~[/tex]
[tex]\bf ~ VII)~\blue{\underline{a =7}} \Rightarrow 7 +b=9\Rightarrow b=9-7\Rightarrow \blue{\underline{b=2}}[/tex]
[tex] \Rightarrow \red{\boxed{\bf \overline{ab}=72}}[/tex]
[tex] \Rightarrow \red{\bf \overline{ba}=27}[/tex]
[tex]\bf ~[/tex]
[tex]\bf ~ VIII)~\blue{\underline{a = 8}} \Rightarrow 8 +b=9\Rightarrow b=9-8\Rightarrow \blue{\underline{b=1}}[/tex]
[tex] \Rightarrow \red{\boxed{\bf \overline{ab}=81}}[/tex]
[tex] \Rightarrow \red{\bf \overline{ba}=18}[/tex]
Numere naturale de forma [tex]\bf\overline{ab}[/tex] sunt:
[tex]\pink{\boxed{\bf \overline{ab}\in \big\{18;27;36;45;54;63;72;81\big\}}}[/tex]
==pav38==
Baftă multă !