Răspuns: 6 numere
[tex]\color{red}\boxed{\boxed{\bf\overline{abcd}\in\{5928, 2531, 5452, 1351, 2362,7391\}}}[/tex]
Explicație pas cu pas:
Salutare!
[tex]\bf \overline{abcd} = ???[/tex]
[tex]\bf a,b,c,d - cifre[/tex]
[tex]\bf a\neq 0[/tex]
[tex]\bf \overline{abc}= \overline{d0}+c^{b}[/tex]
[tex]\bf \overline{abcd}= \overline{abc}\cdot 10+d[/tex]
[tex]\bf Inlocuim~pe ~\overline{abc}~si~vom~avea:~ \overline{abcd}= (\overline{d0}+c^{b})\cdot 10+d[/tex]
[tex]\bf \overline{abcd}= \overline{d00}+10c^{b}+d[/tex]
[tex]\text{\it Descompunem in baza 10}[/tex]
[tex]\bf 1000a +100b+10c+d= \overline{d00}+10c^{b}+d[/tex]
[tex]\bf 1000a +100b+10c= \overline{d00}+10c^{b}+d-d[/tex]
[tex]\bf 1000a +100b+10c= \overline{d00}+10c^{b}[/tex]
[tex]\bf 1000a +100b+10c= 100d+10c^{b}~~\bigg|:10[/tex]
[tex]\bf 100a +10b+c= 10d+c^{b}[/tex]
[tex]\bf c^{b} =100a +10b+c- 10d[/tex]
[tex]\bf c^{b} =100a +10\cdot(b-d) +c[/tex]
[tex]\bf c^{b} va~ fi~ un~ numar~ format~ din~ 3~ cifre~care~adunat ~cu~ \overline{d0} = nr.~ de~ 4~ cifre[/tex]
De acum trebuie sa te dai valori lui c si b astfel incat c^b sa fie un numar de trei cifre. Nu voi lua toate cazurile care nu convin ca e foarte mult de scris
[tex]\bf \star~~ Daca~ c^{b} = 2^{8} = 256\implies 256=100a +80+2-10d[/tex]
[tex]\bf \implies 174 = 100a -10d \implies 87 = 50-5d~~nu ~convine[/tex]
[tex]\bf \star~~ Daca~ c^{b} = 2^{9} = 512\implies 512=100a +90+2-10d[/tex]
[tex]\bf 420=100a -10d\implies d = 10a-42\implies a= 5; d =8;c =2;b=9[/tex]
[tex]\bf \overline{abcd} = 5928~~solutie[/tex]
[tex]\bf \star~~ Daca~ c^{b} = 3^{5} = 243\implies 243=100a +50+3-10d[/tex]
[tex]\bf \implies 190=100a -10d \implies d = 10a- 19\implies a =2; b=5;c=3;d=1[/tex]
[tex]\bf \overline{abcd} = 2531~~solutie[/tex]
[tex]\bf \star~~ Daca~ c^{b} = 5^{3} = 125\implies 125=100a +30+5-10d\implies[/tex]
[tex]\bf\implies 90=100a -10d\implies d=10-9\implies a=1;b=3;c=5;d=1[/tex]
[tex]\bf \overline{abcd} = 1351~~solutie[/tex]
[tex]\bf \star~~ Daca~ c^{b} = 5^{4} = 625\implies 125=100a +40+5-10d\implies[/tex]
[tex]\bf\implies 580=100a -10d\implies d=10a-58\implies a=5;b=4;c=5;d=2[/tex]
[tex]\bf \overline{abcd} = 5452~~solutie[/tex]
[tex]\bf \star~~ Daca~ c^{b} = 6^{3} = 216\implies 216=100a +30+6-10d\implies[/tex]
[tex]\bf \implies 180=100a -10d\implies d = 10a -18\implies a=2;b=3;c=6;d=2[/tex]
[tex]\bf \overline{abcd} = 2362~~solutie[/tex]
[tex]\bf \star~~ Daca~ c^{b} = 9^{3} = 7292\implies 729=100a +30+9-10d \implies[/tex]
[tex]\bf \implies 690=100a -10d\implies d = 10a -69\implies a=7;b=3;c=9;d=1[/tex]
[tex]\bf \overline{abcd} = 7391~~solutie[/tex]
[tex]\color{red}\boxed{\boxed{\text{\bf Din cazurile analizate avem:}~\bf\overline{abcd}\in\{5928, 2531, 5452, 1351, 2362,7391\}}}[/tex]PS: Daca esti pe telefon te rog sa glisezi spre dreapta pentru a vedea rezolvarea completa