Question

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

Answer 1

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

Explicație pas cu pas:

$\bf ~$

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

a, b, c = cifre

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

$\bf a\neq 0$

$\bf ~$

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

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

$\bf ~$

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

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

$\bf ~$

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

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

$\bf ~$

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

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

$\bf~$

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

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

$\bf~$

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

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

$\bf ~$

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

==pav38==

Baftă multă !