Question

x=1x2x3x4x...........x40
a)aratati ca x se divide cu 5^9
b)determinati cea mai mare valoare a lu n, astfel in cat 3^n il divide pe x

Answer 1

X=1*2*3*4*5*...*10*...*15*...*20*...*25*...*30*...*35*...*40*
X=1*2*3*4*5*...2*5*...*3*5*...*4*5...*5*5*...6*5*...*7*5*...8*5
X=5*5*5*5*5*5*5*5*5*(1*2*3*4*....39*8 - Factorii de 5)
X=5^9*(1*2*3*4*....*39*8 - Factorii de 5 )

b) x=1*2*3*...*6*...*9*...*12*...(evidentiezi asa toti multiplii de 3, ei sunt 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 si 39)....*40

x=1*2*3*...*2*3*...3*3*...*4*3...(evidentiezi mai departe multiplii lui 3 descompusi in 3inmultit cu..., nota: 18=3*3*2, 27=3*3*3, 36=3*3*4)

x=3^18*(1*2*4*5*....*40 - Factorii de 3)

Cea mai mare valoare a lui n este 18