Question

Un numar de forma abc se numeste "maret" daca b =a^c. Cate numere marete exista?

Answer 1

Răspuns:$\color{Teal}\boxed{\huge\bf 29~ numere~marete}$

Explicație pas cu pas:

(*^▽^*) Buna!

$\bf Fie ~\overline{abc}- numerele~ marete,~daca ~ b = a^{c}$

$\bf a, b, c - cifre$

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

$\bf a\neq 0$                                                                                                                

$\bf \star Daca ~\underline{a = 1} \implies c~poate~lua~orice~valoare~de~ la~ 0~ pana~ la~ 9$

$\bf 1^{0} =1;~1^{1} =1;~1^{2} =1;~1^{3} =1;.....1^{9} = 1 \implies$

$\bf \implies b = 1\implies \overline{abc}\in\{110,111,112,113,114,115,116,117,118,119\}$

$\bf \star Daca ~\underline{a = 2} \implies c\in \{0,1,2,3,\}\implies b\in\{2^{0},2^{1},2^{2},2^{3}\} \implies$

$\bf \implies b\in\{1,2,4,8\} \implies \overline{abc}\in\{210,221,242,283\}$

$\bf \star Daca ~\underline{a = 3} \implies c\in \{0,1,2\}\implies b\in\{3^{0},3^{1},3^{2}\} \implies b\in\{1,3,9\}\implies$

$\bf \implies \overline{abc}\in\{310,331,392\}$

$\bf \star Daca ~\underline{a = 4} \implies c\in \{0,1\}\implies b\in\{4^{0},4^{1}\} \implies b\in\{1,4\}\implies$

$\bf \implies \overline{abc}\in\{410,441\}$

$\bf \star Daca ~\underline{a = 5} \implies c\in \{0,1\}\implies b\in\{5^{0},5^{1}\} \implies b\in\{1,5\}\implies$

$\bf \implies \overline{abc}\in\{510,551\}$

$\bf \star Daca ~\underline{a = 6} \implies c\in \{0,1\}\implies b\in\{6^{0},6^{1}\} \implies b\in\{1,6\}\implies$

$\bf \implies \overline{abc}\in\{610,661\}$

$\bf \star Daca ~\underline{a = 7} \implies c\in \{0,1\}\implies b\in\{7^{0},7^{1}\} \implies b\in\{1,7\}\implies$

$\bf \implies \overline{abc}\in\{710,771\}$

$\bf \star Daca ~\underline{a = 8} \implies c\in \{0,1\}\implies b\in\{8^{0},8^{1}\} \implies b\in\{1,8\}\implies$

$\bf \implies \overline{abc}\in\{810,881\}$

$\bf \star Daca ~\underline{a = 9} \implies c\in \{0,1\}\implies b\in\{9^{0},9^{1}\} \implies b\in\{1,9\}\implies$

$\bf \implies \overline{abc}\in\{910,991\}$

$\text{\bf Din cazurile analizate avem:}~\bf 10+ 4+3+2\cdot 6 = \boxed{\boxed{\bf 29~ numere~marete}}$

PS: Te rog sa glisezi spre dreapta pentru a vedea toată rezolvarea

#copaceibrainly