Question

dintre numerele 36^12 și 12^36 mai mulți divizori are numărul ​

Answer 1

 

$\displaystyle\bf\\Explicatie:\\O~putere~a~unui~numar~prim~are~~n+1~divizori,\\unde~n~este~exponentul~puterii.\\\\Exemple:\\3^2~are~divizorii:~~\{3^0;~3^1;~3^2\}\\2^4~are~divizorii:~~\{2^0;~2^1;~2^2;~2^3;~2^4\}\\5^3~are~divizorii:~~\{5^0;~5^1;~5^2;~5^3\}\\\\Rezolvare:$

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$\displaystyle\bf\\Il~descompunem~pe~36^{12}~in~factori~primi.\\\\36=4\times9=2^2\times3^2\\\\36^{12}=\Big(2^2\times3^2\Big)^{12}=\Big(2^2\Big)^{12}\times\Big(3^2\Big)^{12}=2^{24}\times3^{24}\\\\\boxed{\bf36^{12}=2^{24}\times3^{24}~~are~~(24+1)(24+1)=25^2=625~de~divizori}$

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$\displaystyle\bf\\Il~descompunem~pe~12^{36}~in~factori~primi.\\\\12=4\times3=2^2\times3\\\\12^{36}=\Big(2^2\times3\Big)^{36}=\Big(2^2\Big)^{36}\times\Big(3\Big)^{36}=2^{72}\times3^{36}\\\\\boxed{\bf12^{36}=2^{72}\times3^{36}~~are~~(72+1)(36+1)=73\times37=2701~divizori}\\\\\\\implies~\boxed{\boxed{\bf12^{36}~are~mai~multi~divizori~decat~36^{12}}}$