Question

19. Numerele naturale de forma abcd cu cifre distincte pentru care a +d=b+c=5 sunt: 1234.
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Answer 1

Răspuns: $\bf \overline{abcd}\in$ 5230, 5320, 5140, 5410, 3142, 3412, 3502, 3052, 2143, 2413, 2503, 2053, 1234, 1324, 1504, 1054, 4231, 4321, 4051, 4501

Explicație pas cu pas:

Salutare !

$\bf~$

$\bf Fie ~\overline{abcd}~numerele ~cautate$

$\bf a,b,c,d - cifre$

$\bf Cifrele~ sunt: 0,1,2,3,4,5,6,7,8,9$

$\bf a\neq b\neq c\neq d$

$\bf a\neq 0$

$\bf a+d = b+c =5 \Rightarrow \red{a,b,c,d \in\big \{0,1,2,3,4,5\big\}}$

$\bf Caz ~I)~~ \underline{a = 1} \Rightarrow 1 +d = 5\Rightarrow \underline{d = 4}$

$\bf \underline{b = 2} \Rightarrow 2 +c = 5\Rightarrow \underline{c = 3}$

$\bf \underline{b = 3} \Rightarrow 3 +c = 5\Rightarrow \underline{c = 2}$

$\bf \underline{b = 5} \Rightarrow 5 +c = 5\Rightarrow \underline{c = 0}$

$\bf \underline{b = 0} \Rightarrow 0 +c = 5\Rightarrow \underline{c = 5}$

$\pink{\boxed{\bf \overline{abcd}\in \big\{1234,1324,1504,1054\big\}}}$

$\bf~$

$\bf Caz ~II)~~ \underline{a = 2} \Rightarrow 2 +d = 5\Rightarrow \underline{d = 3}$

$\bf \underline{b = 1} \Rightarrow 1 +c = 5\Rightarrow \underline{c = 4}$

$\bf \underline{b = 4} \Rightarrow 4 +c = 5\Rightarrow \underline{c = 1}$

$\bf \underline{b = 5} \Rightarrow 5 +c = 5\Rightarrow \underline{c = 0}$

$\bf \underline{b = 0} \Rightarrow 0 +c = 5\Rightarrow \underline{c = 5}$

$\purple{\boxed{\bf \overline{abcd}\in \big\{2143,2413,2503,2053\big\}}}$

$\bf ~$

$\bf Caz ~III)~~ \underline{a = 3} \Rightarrow 3 +d = 5\Rightarrow \underline{d = 2}$

$\bf \underline{b = 1} \Rightarrow 1 +c = 5\Rightarrow \underline{c = 4}$

$\bf \underline{b = 4} \Rightarrow 4 +c = 5\Rightarrow \underline{c = 1}$

$\bf \underline{b = 5} \Rightarrow 5 +c = 5\Rightarrow \underline{c = 0}$

$\bf \underline{b = 0} \Rightarrow 0 +c = 5\Rightarrow \underline{c = 5}$

$\blue{\boxed{\bf \overline{abcd}\in \big\{3142,3412,3502,3052\big\}}}$

$\bf~$

$\bf Caz ~IV)~~ \underline{a = 4} \Rightarrow 4 +d = 5\Rightarrow \underline{d = 1}$

$\bf \underline{b = 2} \Rightarrow 2 +c = 5\Rightarrow \underline{c = 3}$

$\bf \underline{b = 3} \Rightarrow 3 +c = 5\Rightarrow \underline{c = 2}$

$\bf \underline{b = 5} \Rightarrow 5 +c = 5\Rightarrow \underline{c = 0}$

$\bf \underline{b = 0} \Rightarrow 0 +c = 5\Rightarrow \underline{c = 5}$

$\green{\boxed{\bf \overline{abcd}\in \big\{4231,4321,4051,4501\big\}}}$

$\bf~$

$\bf Caz ~V)~~ \underline{a = 5} \Rightarrow 5 +d = 5\Rightarrow \underline{d = 0}$

$\bf \underline{b = 2} \Rightarrow 2 +c = 5\Rightarrow \underline{c = 3}$

$\bf \underline{b = 3} \Rightarrow 3 +c = 5\Rightarrow \underline{c = 2}$

$\bf \underline{b = 1} \Rightarrow 1 +c = 5\Rightarrow \underline{c = 4}$

$\bf \underline{b = 4} \Rightarrow 4 +c = 5\Rightarrow \underline{c = 1}$

$\red{\boxed{\bf \overline{abcd}\in \big\{5230,5320,5140,5410\big\}}}$

$\bf~$

$\bf Din~cazurile ~analizate~ nr.~ naturale~ce~ respecta~ conditiile~ problemei:$

$\bf \overline{abcd}\in$ 5230, 5320, 5140, 5410, 3142, 3412, 3502, 3052, 2143, 2413, 2503, 2053, 1234, 1324, 1504, 1054, 4231, 4321, 4051, 4501

==pav38==

Baftă multă !