Question

Determinati numerele naturale de forma abc stiind ca a+2b +c=6

Answer 1

Răspuns: $\pink{\boxed{\bf \overline{abc}\in \big\{105;113;121;204;212;220;303;311;402;410;501;600\big\}}}$

Explicație pas cu pas:

$\bf ~$

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

a, b, c = cifre

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

$\bf a\neq 0$

Analizăm în funcție de ce valoare poate lua a

$\bf ~$

$\bf ~ I)~\blue{\underline{a = 1}} \Rightarrow 1 +2b+c=6\Rightarrow 2b+c=6-1\Rightarrow \underline{2b+c= 5}$

• $\bf Daca~\underline{b = 0} \Rightarrow2\cdot0+c= 5 \Rightarrow\underline{c = 5} \Rightarrow \red{\boxed{\bf \overline{abc}=105}}$
• $\bf Daca~\underline{b = 1} \Rightarrow2\cdot1+c= 5 \Rightarrow\underline{c = 3} \Rightarrow \red{\boxed{\bf \overline{abc}=113}}$
• $\bf Daca~\underline{b = 2} \Rightarrow2\cdot2+c= 5 \Rightarrow\underline{c = 1} \Rightarrow \red{\boxed{\bf \overline{abc}=121}}$

$\bf ~$

$\bf ~ II)~ \blue{\underline{a = 2}} \Rightarrow 2 +2b+c=6\Rightarrow \underline{2b+c= 4}$

• $\bf Daca~\underline{b = 0} \Rightarrow2\cdot0+c= 4 \Rightarrow\underline{c = 4} \Rightarrow \red{\boxed{\bf \overline{abc}=204}}$
• $\bf Daca~\underline{b = 1} \Rightarrow2\cdot1+c= 4 \Rightarrow\underline{c = 2} \Rightarrow \red{\boxed{\bf \overline{abc}=212}}$
• $\bf Daca~\underline{b = 2} \Rightarrow2\cdot2+c= 4 \Rightarrow\underline{c = 0} \Rightarrow \red{\boxed{\bf \overline{abc}=220}}$

$\bf ~$

$\bf ~ III)~\blue{\underline{a =3}} \Rightarrow 3 +2b+c=6\Rightarrow 2b+c=6-3\Rightarrow \underline{2b+c= 3}$

• $\bf Daca~\underline{b = 0} \Rightarrow2\cdot0+c= 3 \Rightarrow\underline{c =3} \Rightarrow \red{\boxed{\bf \overline{abc}=303}}$
• $\bf Daca~\underline{b = 1} \Rightarrow2\cdot1+c= 3 \Rightarrow\underline{c = 1} \Rightarrow \red{\boxed{\bf \overline{abc}=311}}$

$\bf ~$

$\bf ~ IV)~ \blue{\underline{a =4}} \Rightarrow 4 +2b+c=6\Rightarrow 2b+c=6-4\Rightarrow \underline{2b+c= 2}$

• $\bf Daca~\underline{b = 0} \Rightarrow2\cdot0+c= 2 \Rightarrow\underline{c =2} \Rightarrow \red{\boxed{\bf \overline{abc}=402}}$
• $\bf Daca~\underline{b = 1} \Rightarrow2\cdot1+c= 2 \Rightarrow\underline{c = 0} \Rightarrow \red{\boxed{\bf \overline{abc}=410}}$

$\bf~$

$\bf ~ V)~ \blue{\underline{a =5}} \Rightarrow 5 +2b+c=6\Rightarrow \underline{2b+c= 1}$

• $\bf Daca~\underline{b = 0} \Rightarrow2\cdot0+c= 1 \Rightarrow\underline{c =1} \Rightarrow \red{\boxed{\bf \overline{abc}=501}}$

$\bf~$

$\bf ~ VI)~ \blue{\underline{a =6}} \Rightarrow 6 +2b+c=6\Rightarrow \underline{2b+c=6}$

• $\bf Daca~\underline{b = 0} \Rightarrow2\cdot0+c= 0 \Rightarrow\underline{c =0} \Rightarrow \red{\boxed{\bf \overline{abc}=600}}$

$\bf ~$

Numere naturale sunt ce respectă conditiile problemei sunt:$\pink{\boxed{\bf \overline{abc}\in \big\{105;113;121;204;212;220;303;311;402;410;501;600\big\}}}$

==pav38==

Baftă multă !