Question

Determinati toate nr de forma ab stiind ca ab+ ba = 99
Poate fi a=b? Motivati Repede va rog, Mulțumesc!

Answer 1

Răspuns: $\pink{\boxed{\bf \overline{ab}\in \big\{18;27;36;45;54;63;72;81\big\}}}$

Explicație pas cu pas:

$\bf ~$

$\bf ~\overline{ab}~numerele ~cautate$

a, b = cifre

$\bf a,b\in \{ 0,1,2,3,4,5,6,7,8,9\}$

$\bf a\neq 0~~~b\neq 0$

$\bf \overline{ab}+\overline{ba}= 99$

Descompunem în bază zece avem:

$\bf 10a+b+10b+a=99$

$\bf 11a+11b=99~~\bigg|:11$

$\bf \underline{a+b=9}$  

Analizăm în funcție de ce valoare poate lua a

$\bf ~ I)~\blue{\underline{a = 1}} \Rightarrow 1 +b=9\Rightarrow b=9-1\Rightarrow \blue{\underline{b=8}}$

$\Rightarrow \red{\boxed{\bf \overline{ab}=18}}$

$\Rightarrow \red{\bf \overline{ba}=81}$

$\bf ~$

$\bf ~ II)~\blue{\underline{a = 2}} \Rightarrow 2 +b=9\Rightarrow b=9-2\Rightarrow \blue{\underline{b=7}}$

$\Rightarrow \red{\boxed{\bf \overline{ab}=27}}$

$\Rightarrow \red{\bf \overline{ba}=72}$

$\bf ~$

$\bf ~ III)~\blue{\underline{a = 3}} \Rightarrow 3 +b=9\Rightarrow b=9-3\Rightarrow \blue{\underline{b=6}}$

$\Rightarrow \red{\boxed{\bf \overline{ab}=36}}$

$\Rightarrow \red{\bf \overline{ba}=63}$

$\bf ~$

$\bf ~ IV)~\blue{\underline{a = 4}} \Rightarrow 4 +b=9\Rightarrow b=9-4\Rightarrow \blue{\underline{b=5}}$

$\Rightarrow \red{\boxed{\bf \overline{ab}=45}$

$\Rightarrow \red{\bf \overline{ba}=54}$

$\bf ~$

$\bf ~ V)~\blue{\underline{a = 5}} \Rightarrow 5 +b=9\Rightarrow b=9-5\Rightarrow \underline{b=4}$

$\Rightarrow \red{\boxed{\bf \overline{ab}=54}}$

$\Rightarrow \red{\bf \overline{ba}=45}$

$\bf ~$

$\bf ~ VI)~\blue{\underline{a = 6}} \Rightarrow 6 +b=9\Rightarrow b=9-6\Rightarrow \blue{\underline{b=3}}$

$\Rightarrow \red{\boxed{\bf \overline{ab}=63}}$

$\Rightarrow \red{\bf \overline{ba}=36}$

$\bf ~$

$\bf ~ VII)~\blue{\underline{a =7}} \Rightarrow 7 +b=9\Rightarrow b=9-7\Rightarrow \blue{\underline{b=2}}$

$\Rightarrow \red{\boxed{\bf \overline{ab}=72}}$

$\Rightarrow \red{\bf \overline{ba}=27}$

$\bf ~$

$\bf ~ VIII)~\blue{\underline{a = 8}} \Rightarrow 8 +b=9\Rightarrow b=9-8\Rightarrow \blue{\underline{b=1}}$

$\Rightarrow \red{\boxed{\bf \overline{ab}=81}}$

$\Rightarrow \red{\bf \overline{ba}=18}$

Numere naturale de forma $\bf\overline{ab}$ sunt:

$\pink{\boxed{\bf \overline{ab}\in \big\{18;27;36;45;54;63;72;81\big\}}}$

a ≠ b, deoarece a + b = 9

                                        ↓

                                     IMPAR    

Dacă o sumă de două numere este impară atunci numerele respective nu pot fi egale.  ==pav38==

Baftă multă !