Răspuns :
[tex]\bf Fie ~\overline{abcd}~numerele ~cautate[/tex]
[tex]\bf a,b,c,d~\in \{ 0,1,2,3,4,5,6,7,8,9\}[/tex]
[tex]\bf a\neq b\neq c\neq d[/tex]
[tex]\bf a\neq 0[/tex]
[tex]\bf a+d = b+c =7 \Rightarrow \red{a,b,c,d \in \{0,1,2,3,4,5,6,7\}}[/tex]
[tex]\bf Caz ~I)~ \underline{a = 1} \Rightarrow 1 +d = 7\Rightarrow \underline{d = 6}[/tex]
[tex]\bf \underline{b = 2} \Rightarrow 2 +c = 7\Rightarrow \underline{c = 5}[/tex]
[tex]\bf \underline{b = 3} \Rightarrow 3 +c = 7\Rightarrow \underline{c = 4}[/tex]
[tex]\bf \underline{b = 4} \Rightarrow 4 +c = 7\Rightarrow \underline{c = 3}[/tex]
[tex]\bf \underline{b = 5} \Rightarrow 5 +c = 7\Rightarrow \underline{c = 2}[/tex]
[tex]\bf \underline{b = 7} \Rightarrow 7 +c = 7\Rightarrow \underline{c = 0}[/tex]
[tex]\bf \underline{b = 0} \Rightarrow 0 +c = 7\Rightarrow \underline{c = 7}[/tex]
[tex]\pink{\boxed{\bf \overline{abcd}\in \big\{1256,1346,1436,1526, 1706,1076 \big\}}}[/tex]
[tex]\bf Caz ~II)~~ \underline{a = 2} \Rightarrow 2 +d = 7\Rightarrow \underline{d = 5}[/tex]
[tex]\bf \underline{b = 1} \Rightarrow 1 +c = 7\Rightarrow \underline{c = 6}[/tex]
[tex]\bf \underline{b = 3} \Rightarrow 3 +c = 7\Rightarrow \underline{c = 4}[/tex]
[tex]\bf \underline{b = 4} \Rightarrow 4 +c = 7\Rightarrow \underline{c = 3}[/tex]
[tex]\bf \underline{b = 6} \Rightarrow 6 +c = 7\Rightarrow \underline{c = 1}[/tex]
[tex]\bf \underline{b = 7} \Rightarrow 7 +c = 7\Rightarrow \underline{c = 0}[/tex]
[tex]\bf \underline{b = 0} \Rightarrow 0 +c = 7\Rightarrow \underline{c = 7}[/tex]
[tex]\purple{\boxed{\bf \overline{abcd}\in \big\{2165,2345,2435,2615,2705,2075\big\}}}[/tex]
[tex]\bf Caz ~III)~~ \underline{a = 3} \Rightarrow 3 +d = 7\Rightarrow \underline{d = 4}[/tex]
[tex]\bf \underline{b = 1} \Rightarrow 1 +c = 7\Rightarrow \underline{c = 6}[/tex]
[tex]\bf \underline{b = 2} \Rightarrow 2 +c = 7\Rightarrow \underline{c = 5}[/tex]
[tex]\bf \underline{b = 5} \Rightarrow 5 +c = 7\Rightarrow \underline{c = 2}[/tex]
[tex]\bf \underline{b = 6} \Rightarrow 6 +c = 7\Rightarrow \underline{c = 1}[/tex]
[tex]\bf \underline{b = 7} \Rightarrow 7 +c = 7\Rightarrow \underline{c = 0}[/tex]
[tex]\bf \underline{b = 0} \Rightarrow 0 +c = 7\Rightarrow \underline{c = 7}[/tex]
[tex]\blue{\boxed{\bf \overline{abcd}\in \big\{3164,3254,3524, 3614,3704,3074\big\}}}[/tex]
[tex]\bf Caz ~IV)~~ \underline{a = 4} \Rightarrow 4 +d = 7\Rightarrow \underline{d = 3}[/tex]
[tex]\bf \underline{b = 1} \Rightarrow 1 +c = 7\Rightarrow \underline{c = 6}[/tex]
[tex]\bf \underline{b = 2} \Rightarrow 2 +c = 7\Rightarrow \underline{c = 5}[/tex]
[tex]\bf \underline{b = 5} \Rightarrow 5 +c = 7\Rightarrow \underline{c = 2}[/tex]
[tex]\bf \underline{b = 6} \Rightarrow 6 +c = 7\Rightarrow \underline{c = 1}[/tex]
[tex]\bf \underline{b = 7} \Rightarrow 7 +c = 7\Rightarrow \underline{c = 0}[/tex]
[tex]\bf \underline{b = 0} \Rightarrow 0 +c = 7\Rightarrow \underline{c = 7}[/tex]
[tex]\green{\boxed{\bf \overline{abcd}\in \big\{4163,4253,4523,4613,4703,4073\big\}}}[/tex]
[tex]\bf Caz ~V)~~ \underline{a = 5} \Rightarrow 5 +d = 7\Rightarrow \underline{d = 2}[/tex]
[tex]\bf \underline{b = 1} \Rightarrow 1 +c = 7\Rightarrow \underline{c = 6}[/tex]
[tex]\bf \underline{b = 3} \Rightarrow 3 +c = 7\Rightarrow \underline{c = 4}[/tex]
[tex]\bf \underline{b = 4} \Rightarrow 4 +c = 7\Rightarrow \underline{c = 3}[/tex]
[tex]\bf \underline{b = 6} \Rightarrow 6 +c = 7\Rightarrow \underline{c = 1}[/tex]
[tex]\bf \underline{b = 7} \Rightarrow 7 +c = 7\Rightarrow \underline{c = 0}[/tex]
[tex]\bf \underline{b = 0} \Rightarrow 0 +c = 7\Rightarrow \underline{c = 7}[/tex]
[tex]\orange{\boxed{\bf \overline{abcd}\in \big\{5162,5342,5432,5612,5702,5072\big\}}}[/tex]
[tex]\bf Caz ~VI)~~ \underline{a = 6} \Rightarrow 6 +d = 7\Rightarrow \underline{d = 1}[/tex]
[tex]\bf \underline{b = 2} \Rightarrow 2 +c = 7\Rightarrow \underline{c = 5}[/tex]
[tex]\bf \underline{b = 3} \Rightarrow 3 +c = 7\Rightarrow \underline{c = 4}[/tex]
[tex]\bf \underline{b = 4} \Rightarrow 4 +c = 7\Rightarrow \underline{c = 3}[/tex]
[tex]\bf \underline{b = 5} \Rightarrow 5 +c = 7\Rightarrow \underline{c = 2}[/tex]
[tex]\bf \underline{b = 7} \Rightarrow 7 +c = 7\Rightarrow \underline{c = 0}[/tex]
[tex]\bf \underline{b = 0} \Rightarrow 0 +c = 7\Rightarrow \underline{c = 7}[/tex]
[tex]\pink{\boxed{\bf \overline{abcd}\in \big\{6251,6341,6431,6521, 6701,6071 \big\}}}[/tex]
[tex]\bf Caz ~VII)~~ \underline{a = 7} \Rightarrow 7 +d = 7\Rightarrow \underline{d = 0}[/tex]
[tex]\bf \underline{b = 1} \Rightarrow 1 +c = 7\Rightarrow \underline{c = 6}[/tex]
[tex]\bf \underline{b = 2} \Rightarrow 2 +c = 7\Rightarrow \underline{c = 5}[/tex]
[tex]\bf \underline{b = 3} \Rightarrow 3 +c = 7\Rightarrow \underline{c = 4}[/tex]
[tex]\bf \underline{b = 4} \Rightarrow 4 +c = 7\Rightarrow \underline{c = 3}[/tex]
[tex]\bf \underline{b = 5} \Rightarrow 5 +c = 7\Rightarrow \underline{c = 2}[/tex]
[tex]\bf \underline{b = 6} \Rightarrow 6 +c = 7\Rightarrow \underline{c = 1}[/tex]
[tex]\purple{\boxed{\bf \overline{abcd}\in \big\{7160,7250,7340,7430,7520,7610\big\}}}[/tex]
