Question

Determinati:
A) nr xyz in baza 10, stiind ca xy + yz+ zx= 77 , unde x, y,z cifre distincte
B)Nr naturale n ,pt care : 8la n +8 la n + 1 = 18× 2la 2015

Answer 1

a)
xy + yz+ zx= 77
10x+y+10y+z+10z+x=77
11x+11y+11z=77
11(x+y+z)=77
x+y+z=7

x=1, y=2, z=4,  xyz=124
x=1, y=4, z=2,  xyz=142

x=2, y=1, z=4,  xyz=214
x=2, y=4, z=1,  xyz=241

x=4, y=1, z=2,  xyz=412
x=4, y=2, z=1,  xyz=421

b)
8^n +8^(n+1)=18x2^2015

(8^n)(1+8)=18x2^2015
9x(8^n)=18x2^2015
(8^n)=2x2^2015
(2^3)^n=2^2016
2^(3n)=2^2016
3n=2016

n=672