Question

1) Fie a=7,2(31).Determinati duma primelor 100 de zecimale ale nr a.
2) Determonati toate nr de foarma abc scrise in baza 10,astfel incat abc la bc da catul 5 si restul 4
Multumesc !!

Answer 1

1. Suma primelor 100 zecimale = 2+3+1+3+1*48+3*48 = 9+48+144= 201
2. abc=bc*5+4 ⇔ 100a+10b+c=(10b+c)*5+4 ⇔ 100a+10b+c=50b+5c+4 
⇔100a=40b+4c+4, unde a, b,c sunt cifre si ∈ {0,1,2,3,4,5,6,7,8,9}
daca a = 0 => 40b+4c+4=0 imposibil
daca a=1 => 100=40b+4c+4 => 40b+4c=96 =>10b+c=24 => b=2, c=4
daca a=2 => 40b+4c+4=200 => 40b+4c=196 =>10b+c=49 => b=4, c=9
daca a=3 => 40b+4c+4=300 => 40b+4c=296=> 10b+c=74 => b=7, c=4
daca a=4 => 40b+4c+4=400=> 40b+4c=396 => 10b+c=99 => b=9, c=9
daca a=5 => 40b+4c+4=500 => 40b+4c=496 => 10b+c=124 imposibl
similar cu a=6, a=7, a=8, a=9

Asadar nr de forma abc = 124, 249, 374, 499