Răspuns:
a = 2017 + 2 × (1 + 2 + ... + 2016)
a = 2017 + 2 × 2016 × (2016 + 1) : 2
a = 2017 + 2 × 2016 × 2017 : 2
a = 2017 + 2016 × 2017
a = 2017 × ( 1 + 2016)
a = 2017 × 2017
a = 2017^2 patrat perfect
b=1+3+5+...+2017=
=1+2+3+4+5+6+...+2016+2017-(2+4+6+...+2016)=
=2017×2018:2-2(1+2+3+...+1008)=
=2017×1009-2×1008×1009:2=
=2017×1009-1008×1009=
=1009×(2017-1008)=
=1009×1009=
=1009²
c= 81(1+2+3+.....+49)
c=81•(49•50):2
c=81•49•25
c=9^2•7^2•5^2
c=(9•7•5)^2
c=315^2 (patrat perfect)
d = 2 × (1 + 2 + 3 + ... + 124) + 125
d = 2 × 124 × (124 + 1) : 2 + 125
d = 2 × 124 × 125 : 2 + 124
am folosit suma lui Gauss
2: 2 = 1 si de aceea se reduc
d = 124 × 125 + 125
dam factor comun pe 125
d = 125 × ( 124 + 1)
d = 125 × 125
d = 125^2
d = (5^3)^2
d = (5^2)^3
d = 25^3 cub perfect
