Question

Determinati cel mai mare numar natural n, astfel incat 2^n | ( 2 x 4 x 6 x ... x 50 )

Answer 1

$2*4*6*8*....*50= \\ (2*1)*(2*2)*(2*3)*(2*4)*.....*(2*25) \\ 2^{25}*1*2*3*4*5*6*...*25.$
Puterea la care se gaseste 2 in produsul $1*2*3*4*5*6*7*......*25$ se afla dupa formula: $[\frac{25}{2}]+[\frac{25}{2^{2}}]+[\frac{25}{2^{3}}]+[\frac{25}{2^{4}}]+[\frac{25}{2^{5}}]=12+6+3+1+0=22$
[] reprezinta partea intreaga
Deci n=25+22=47