Question

Efectuati
a) (1+2+3+...+20) : 21=?
b) (11+11*2+11*2^2+...+11*2^9) :1023=?
c) (5^3*10+5^3*11+5^3*12+...+5^3*100) :1001=?
d) [20+2*(1+2+3+...+19)] : 10^2 - 2^2=?
Aflati x nr natural stiind ca
{10^3+10^2*[10+(5^2-x) : 5]} :20^2=6
(11+11^2+11^3) :133+x :2^4=?
Aflati x din
99+2(1+2+3+...+x)=99^2
Va rog frumos ajutatima!!!

Answer 1

a) (1+2+3+...+20) : 21==
=20*21:2:21=10

b) (11+11*2+11*2^2+...+11*2^9) :1023=
=11(1+2+2^2+...+2^9):1023=
=11(1+(2^10-2)/(2-1):1023=
=11(1+2*10-2):1023=
=(2*10-1):93=
=(1024-1):93=
=1023:93=11

c) (5^3*10+5^3*11+5^3*12+...+5^3*100) :1001=
= 5^3(10+11+12+...+100) :1001=
= 5^3*(10*91+90*91/2) :1001=
= 5^3*(910+4095) :1001=
= 5^3*5005 :1001=
= 5^3*5=
= 5^4

d) [20+2*(1+2+3+...+19)] : 10^2 - 2^2=
= [20+2*19*20:2] : 10^2 - 2^2=
= [20+19*20] : 10^2 - 2^2=
= [20(1+19)] : 10^2 - 2^2=
= 20*20 : 10^2 - 2^2=
=2^2 * 10^2:10^2 - 2^2
=2^2-2^2=0

Aflati x nr natural stiind ca
{10^3+10^2*[10+(5^2-x) : 5]} :20^2=6
10^3+10^2*[10+(5^2-x) : 5] =2^4×5^2 × 2 ×3
10^2*[10+(5^2-x) : 5] =2^5×5^2 ×3-2^3×5^3
10^2*[10+(5^2-x) : 5] =2^3×5^2(2^2×3-5)
2^2*5^2[10+(5^2-x) : 5] =2^3×5^2×7 
10+(5^2-x) : 5=2x7
(5^2-x) : 5=14-10
(5^2-x) : 5=4
25-x =20
x=5

(11+11^2+11^3) :133+x =2^4
(11^4-11)/(11-1) :133+x=2^4
1463:133+x=2^4
11+x=2^4
x=2^4-11
x=16-11
x=5


Aflati x din 
99+2(1+2+3+...+x)=99^2
99+2*x(x+1)/2=99^2
x(x+1)=99^2-99
x(x+1)=99(99-1)
x(x+1)=99*98
x=98