Question

Scrieți toate numerele naturale de forma abcd (cu bara deasupra) cu cifre distincte astfel încât:
a+d=b+c=7

Answer 1

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

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

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

$\bf a\neq 0$

$\bf a+d = b+c =7 \Rightarrow \red{a,b,c,d \in \{0,1,2,3,4,5,6,7\}}$

$\bf Caz ~I)~ \underline{a = 1} \Rightarrow 1 +d = 7\Rightarrow \underline{d = 6}$

$\bf \underline{b = 2} \Rightarrow 2 +c = 7\Rightarrow \underline{c = 5}$

$\bf \underline{b = 3} \Rightarrow 3 +c = 7\Rightarrow \underline{c = 4}$

$\bf \underline{b = 4} \Rightarrow 4 +c = 7\Rightarrow \underline{c = 3}$

$\bf \underline{b = 5} \Rightarrow 5 +c = 7\Rightarrow \underline{c = 2}$

$\bf \underline{b = 7} \Rightarrow 7 +c = 7\Rightarrow \underline{c = 0}$

$\bf \underline{b = 0} \Rightarrow 0 +c = 7\Rightarrow \underline{c = 7}$

$\pink{\boxed{\bf \overline{abcd}\in \big\{1256,1346,1436,1526, 1706,1076 \big\}}}$

$\bf Caz ~II)~~ \underline{a = 2} \Rightarrow 2 +d = 7\Rightarrow \underline{d = 5}$

$\bf \underline{b = 1} \Rightarrow 1 +c = 7\Rightarrow \underline{c = 6}$

$\bf \underline{b = 3} \Rightarrow 3 +c = 7\Rightarrow \underline{c = 4}$

$\bf \underline{b = 4} \Rightarrow 4 +c = 7\Rightarrow \underline{c = 3}$

$\bf \underline{b = 6} \Rightarrow 6 +c = 7\Rightarrow \underline{c = 1}$

$\bf \underline{b = 7} \Rightarrow 7 +c = 7\Rightarrow \underline{c = 0}$

$\bf \underline{b = 0} \Rightarrow 0 +c = 7\Rightarrow \underline{c = 7}$

$\purple{\boxed{\bf \overline{abcd}\in \big\{2165,2345,2435,2615,2705,2075\big\}}}$

$\bf Caz ~III)~~ \underline{a = 3} \Rightarrow 3 +d = 7\Rightarrow \underline{d = 4}$

$\bf \underline{b = 1} \Rightarrow 1 +c = 7\Rightarrow \underline{c = 6}$

$\bf \underline{b = 2} \Rightarrow 2 +c = 7\Rightarrow \underline{c = 5}$

$\bf \underline{b = 5} \Rightarrow 5 +c = 7\Rightarrow \underline{c = 2}$

$\bf \underline{b = 6} \Rightarrow 6 +c = 7\Rightarrow \underline{c = 1}$

$\bf \underline{b = 7} \Rightarrow 7 +c = 7\Rightarrow \underline{c = 0}$

$\bf \underline{b = 0} \Rightarrow 0 +c = 7\Rightarrow \underline{c = 7}$

$\blue{\boxed{\bf \overline{abcd}\in \big\{3164,3254,3524, 3614,3704,3074\big\}}}$

$\bf Caz ~IV)~~ \underline{a = 4} \Rightarrow 4 +d = 7\Rightarrow \underline{d = 3}$

$\bf \underline{b = 1} \Rightarrow 1 +c = 7\Rightarrow \underline{c = 6}$

$\bf \underline{b = 2} \Rightarrow 2 +c = 7\Rightarrow \underline{c = 5}$

$\bf \underline{b = 5} \Rightarrow 5 +c = 7\Rightarrow \underline{c = 2}$

$\bf \underline{b = 6} \Rightarrow 6 +c = 7\Rightarrow \underline{c = 1}$

$\bf \underline{b = 7} \Rightarrow 7 +c = 7\Rightarrow \underline{c = 0}$

$\bf \underline{b = 0} \Rightarrow 0 +c = 7\Rightarrow \underline{c = 7}$

$\green{\boxed{\bf \overline{abcd}\in \big\{4163,4253,4523,4613,4703,4073\big\}}}$

$\bf Caz ~V)~~ \underline{a = 5} \Rightarrow 5 +d = 7\Rightarrow \underline{d = 2}$

$\bf \underline{b = 1} \Rightarrow 1 +c = 7\Rightarrow \underline{c = 6}$

$\bf \underline{b = 3} \Rightarrow 3 +c = 7\Rightarrow \underline{c = 4}$

$\bf \underline{b = 4} \Rightarrow 4 +c = 7\Rightarrow \underline{c = 3}$

$\bf \underline{b = 6} \Rightarrow 6 +c = 7\Rightarrow \underline{c = 1}$

$\bf \underline{b = 7} \Rightarrow 7 +c = 7\Rightarrow \underline{c = 0}$

$\bf \underline{b = 0} \Rightarrow 0 +c = 7\Rightarrow \underline{c = 7}$

$\orange{\boxed{\bf \overline{abcd}\in \big\{5162,5342,5432,5612,5702,5072\big\}}}$

$\bf Caz ~VI)~~ \underline{a = 6} \Rightarrow 6 +d = 7\Rightarrow \underline{d = 1}$

$\bf \underline{b = 2} \Rightarrow 2 +c = 7\Rightarrow \underline{c = 5}$

$\bf \underline{b = 3} \Rightarrow 3 +c = 7\Rightarrow \underline{c = 4}$

$\bf \underline{b = 4} \Rightarrow 4 +c = 7\Rightarrow \underline{c = 3}$

$\bf \underline{b = 5} \Rightarrow 5 +c = 7\Rightarrow \underline{c = 2}$

$\bf \underline{b = 7} \Rightarrow 7 +c = 7\Rightarrow \underline{c = 0}$

$\bf \underline{b = 0} \Rightarrow 0 +c = 7\Rightarrow \underline{c = 7}$

$\pink{\boxed{\bf \overline{abcd}\in \big\{6251,6341,6431,6521, 6701,6071 \big\}}}$

$\bf Caz ~VII)~~ \underline{a = 7} \Rightarrow 7 +d = 7\Rightarrow \underline{d = 0}$

$\bf \underline{b = 1} \Rightarrow 1 +c = 7\Rightarrow \underline{c = 6}$

$\bf \underline{b = 2} \Rightarrow 2 +c = 7\Rightarrow \underline{c = 5}$

$\bf \underline{b = 3} \Rightarrow 3 +c = 7\Rightarrow \underline{c = 4}$

$\bf \underline{b = 4} \Rightarrow 4 +c = 7\Rightarrow \underline{c = 3}$

$\bf \underline{b = 5} \Rightarrow 5 +c = 7\Rightarrow \underline{c = 2}$

$\bf \underline{b = 6} \Rightarrow 6 +c = 7\Rightarrow \underline{c = 1}$

$\purple{\boxed{\bf \overline{abcd}\in \big\{7160,7250,7340,7430,7520,7610\big\}}}$