Question

Scrieți toate numerele naturale de forma abcd cu cifre distincte a.î. A+d b+c=7

Answer 1

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

$\bf a,b,c,d ~\leq~9$

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

$\bf a\neq 0$

$\bf a+d = b+c =7 \Rightarrow \red{a,b,c,d \leq 7}$

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

• $\bf \underline{b = 2} \Rightarrow \underline{c = 5}$
• $\bf \underline{b = 3} \Rightarrow \underline{c = 4}$
• $\bf \underline{b = 4} \Rightarrow \underline{c = 3}$
• $\bf \underline{b = 5} \Rightarrow \underline{c = 2}$
• $\bf \underline{b = 7} \Rightarrow \underline{c = 0}$
• $\bf \underline{b = 0} \Rightarrow \underline{c = 7}$

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

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

• $\bf \underline{b = 1}\Rightarrow \underline{c = 6}$
• $\bf \underline{b = 3} \Rightarrow \underline{c = 4}$
• $\bf \underline{b = 4} \Rightarrow \underline{c = 3}$
• $\bf \underline{b = 6} \Rightarrow \underline{c = 1}$
• $\bf \underline{b = 7} \Rightarrow \underline{c = 0}$
• $\bf \underline{b = 0}\Rightarrow \underline{c = 7}$

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

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

• $\bf \underline{b = 1} \Rightarrow \underline{c = 6}$
• $\bf \underline{b = 2} \Rightarrow \underline{c = 5}$
• $\bf \underline{b = 5} \Rightarrow \underline{c = 2}$
• $\bf \underline{b = 6} \Rightarrow \underline{c = 1}$
• $\bf \underline{b = 7} \Rightarrow \underline{c = 0}$
• $\bf \underline{b = 0} \Rightarrow \underline{c = 7}$

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

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

• $\bf \underline{b = 1} \Rightarrow \underline{c = 6}$
• $\bf \underline{b = 2} \Rightarrow \underline{c = 5}$
• $\bf \underline{b = 5} \Rightarrow \underline{c = 2}$
• $\bf \underline{b = 6} \Rightarrow \underline{c = 1}$
• $\bf \underline{b = 7} \Rightarrow \underline{c = 0}$
• $\bf \underline{b = 0} \Rightarrow \underline{c = 7}$

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

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

• $\bf \underline{b = 1} \Rightarrow \underline{c = 6}$
• $\bf \underline{b = 3} \Rightarrow \underline{c = 4}$
• $\bf \underline{b = 4} \Rightarrow \underline{c = 3}$
• $\bf \underline{b = 6} \Rightarrow \underline{c = 1}$
• $\bf \underline{b = 7} \Rightarrow \underline{c = 0}$
• $\bf \underline{b = 0}\Rightarrow \underline{c = 7}$

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

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

• $\bf \underline{b = 2} \Rightarrow \underline{c = 5}$
• $\bf \underline{b = 3} \Rightarrow \underline{c = 4}$
• $\bf \underline{b = 4} \Rightarrow \underline{c = 3}$
• $\bf \underline{b = 5} \Rightarrow \underline{c = 2}$
• $\bf \underline{b = 7} \Rightarrow \underline{c = 0}$
• $\bf \underline{b = 0} \Rightarrow \underline{c = 7}$

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

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

• $\bf \underline{b = 1} \Rightarrow \underline{c = 6}$
• $\bf \underline{b = 2}\Rightarrow \underline{c = 5}$
• $\bf \underline{b = 3} \Rightarrow \underline{c = 4}$
• $\bf \underline{b = 4} \Rightarrow \underline{c = 3}$
• $\bf \underline{b = 5} \Rightarrow \underline{c = 2}$
• $\bf \underline{b = 6} \Rightarrow \underline{c = 1}$

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

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

$\bf \overline{abcd}\in$ 1256, 1346, 1436, 1526, 1706, 1076,  2165, 2345, 2435, 2615, 2705, 2075, 3164, 3254, 3524, 3614, 3704, 3074, 4163, 4253, 4523, 4613, 4703, 4073, 5162, 5342, 5432, 5612, 5702, 5072, 6251, 6341, 6431, 6521, 6701, 6071, 7160, 7250, 7340, 7430, 7520, 7610