Question

Determinati cel mai mare numar natural n,astfel incat:
$2^{n}|(2*4*6*...50)$

Answer 1

$2 \cdot 4 \cdot 6 \cdot...\cdot 50= \\ \\ =(2 \cdot 1 ) (2 \cdot 2 )(2 \cdot 3) ...(2 \cdot 25)= \\ \\ =2^{25} \cdot 1 \cdot 2 \cdot 3 \cdot... \cdot 25. \\ \\ Puterea~la~care~se~gaseste~2~in~produsul~1 \cdot 2 \cdot 3 \cdot ... \cdot 25~se~afla \\ \\dupa~o~formula:~ \Big[\frac{25}{2} \Big ]+ \Big[\frac{25}{2^2} \Big]+ \Big[ \frac{25}{2^3}\Big]+ \Big[\frac{25}{2^4}\Big]=12+6+3+1=22. \\ \\ Notatia ~[x]~reprezinta~partea~intreaga~a~lui~x. \\ \\ Deci~n=25+22=47 .$