Răspuns :
Răspuns:[tex]\color{Teal}\boxed{\huge\bf 29~ numere~marete}[/tex]
Explicație pas cu pas:
(*^▽^*) Buna!
[tex]\bf Fie ~\overline{abc}- numerele~ marete,~daca ~ b = a^{c}[/tex]
[tex]\bf a, b, c - cifre[/tex]
[tex]\bf a, b,c \in \{0,1,2,3,4,5,6,7,8,9\}[/tex]
[tex]\bf a\neq 0[/tex]
[tex]\bf \star Daca ~\underline{a = 1} \implies c~poate~lua~orice~valoare~de~ la~ 0~ pana~ la~ 9[/tex]
[tex]\bf 1^{0} =1;~1^{1} =1;~1^{2} =1;~1^{3} =1;.....1^{9} = 1 \implies[/tex]
[tex]\bf \implies b = 1\implies \overline{abc}\in\{110,111,112,113,114,115,116,117,118,119\}[/tex]
[tex]\bf \star Daca ~\underline{a = 2} \implies c\in \{0,1,2,3,\}\implies b\in\{2^{0},2^{1},2^{2},2^{3}\} \implies[/tex]
[tex]\bf \implies b\in\{1,2,4,8\} \implies \overline{abc}\in\{210,221,242,283\}[/tex]
[tex]\bf \star Daca ~\underline{a = 3} \implies c\in \{0,1,2\}\implies b\in\{3^{0},3^{1},3^{2}\} \implies b\in\{1,3,9\}\implies[/tex]
[tex]\bf \implies \overline{abc}\in\{310,331,392\}[/tex]
[tex]\bf \star Daca ~\underline{a = 4} \implies c\in \{0,1\}\implies b\in\{4^{0},4^{1}\} \implies b\in\{1,4\}\implies[/tex]
[tex]\bf \implies \overline{abc}\in\{410,441\}[/tex]
[tex]\bf \star Daca ~\underline{a = 5} \implies c\in \{0,1\}\implies b\in\{5^{0},5^{1}\} \implies b\in\{1,5\}\implies[/tex]
[tex]\bf \implies \overline{abc}\in\{510,551\}[/tex]
[tex]\bf \star Daca ~\underline{a = 6} \implies c\in \{0,1\}\implies b\in\{6^{0},6^{1}\} \implies b\in\{1,6\}\implies[/tex]
[tex]\bf \implies \overline{abc}\in\{610,661\}[/tex]
[tex]\bf \star Daca ~\underline{a = 7} \implies c\in \{0,1\}\implies b\in\{7^{0},7^{1}\} \implies b\in\{1,7\}\implies[/tex]
[tex]\bf \implies \overline{abc}\in\{710,771\}[/tex]
[tex]\bf \star Daca ~\underline{a = 8} \implies c\in \{0,1\}\implies b\in\{8^{0},8^{1}\} \implies b\in\{1,8\}\implies[/tex]
[tex]\bf \implies \overline{abc}\in\{810,881\}[/tex]
[tex]\bf \star Daca ~\underline{a = 9} \implies c\in \{0,1\}\implies b\in\{9^{0},9^{1}\} \implies b\in\{1,9\}\implies[/tex]
[tex]\bf \implies \overline{abc}\in\{910,991\}[/tex]
[tex]\text{\bf Din cazurile analizate avem:}~\bf 10+ 4+3+2\cdot 6 = \boxed{\boxed{\bf 29~ numere~marete}}[/tex]
PS: Te rog sa glisezi spre dreapta pentru a vedea toată rezolvarea