Question

1.Aratati ca numarul a=(2+4+6+...+2000)*1001^2000 este cub perfect.
2.Determinati x si y stiind ca 4x+5y este patrat perfect si 4x,respectiv 5y sunt numere naturale de 2 cifre scrise in baza 10.

Answer 1

2(1+2+3+.................+1000)*1001²⁰⁰⁰
2*1000*1001/2*1001²⁰⁰⁰=1000*1001²⁰⁰¹=10³*(1001⁶⁶⁷)³=(10*1001⁶⁶⁷)³

4*10+x+5*10+y=90+(x+y)
patrate perfecte 81,100,121........
val.minima 4x+5y=90
val max.     4x+5y=100
x+y=10 x∈{1,2,3,........,9}
             y∈{9,8,7,........,1}