Question

Care este ultima cifra a nr: 1^1×2^2×3^3×4^4÷......×8^8×9^9​

Answer 1

 

$\displaystyle\bf\\Avem~produsul:\\\\1^1\times2^2\times3^3\times4^4\times...\times8^8\times9^9\\\\Scriem~produsul~cu~toti~factorii:\\\\1^1\times2^2\times3^3\times4^4\times5^5\times6^6\times7^7\times8^8\times9^9\\\\\textbf{Printre factorii produsului il gasim pe }~5^5~si~pe~4^4\\\\5^5=5\times5\times5\times5\times5\\\\4^4=\Big(2^2\Big)^4=2^{2\times4}=2^8\\\\2^8=2\times2\times2\times2\times2\times2\times2\times2$

.

$\displaystyle\bf\\Fiecare~5~din~5^5~se~inmulteste~cu~un~2~din~2^8~si rezulta:\\\\(5\times2)\times(5\times2)\times(5\times2)\times(5\times2)\times(5\times2)=\\\\=10\times10\times10\times10\times10=10^5=100000\\\\10^5~adica~100000~il~gasim~printre~factorii~produsului.\\\\\textbf{1~urmat de 0 sau de mai multi de 0, care se inmulteste cu un numar}\\\\\textbf{va face ca ultimele cifre ale rezultatului inmultirii sa fie zerouri.}\\\\\boxed{\textbf{Rezulta ca ultimile 5 cifre ale produsului sunt zerouri.}}$