Question-110887

Question Number 110887 by I want to learn more last updated on 31/Aug/20

Commented by Aina Samuel Temidayo last updated on 31/Aug/20

\exists 100 a + 10 b + c \in (100, 999) s . t .

100 a + 10 b + c = a^{3} + b^{3} + c^{3}

100 a + 10 b + c =

(a + b + c) (a^{2} + b^{2} + c^{2} - ab - bc - ca)

+ 3 abc = 10 (10 a + b) + c

Satisfying these conditions, the

numbers are .

153

370

371

407

They are called ARMSTRONG

numbers .

Commented by I want to learn more last updated on 31/Aug/20

Thanks sir, i appreciate

Answered by Rasheed.Sindhi last updated on 02/Sep/20

100 h + 10 t + u = h^{3} + t^{3} + u^{3}

(h^{3} - 100 h) + (t^{3} - 10 t) + (u^{3} - u) = 0

h (h - 10) (h + 10)

+ t (t^{2} - 10)

+ u (u - 1) (u + 1) = 0 . . A

^{∙} F o r u = 0, 1 :

h (h - 10) (h + 10) + t (t^{2} - 10) = 0

h (10 - h) (10 + h) = t (t^{2} - 10)

h (10 - h) (10 + h) > 0

\Rightarrow t (t^{2} - 10) > 0

\Rightarrow t > 3

u = 0, 1 \Rightarrow t > 3

h^{3} - 100 h + t (t^{2} - 10)

v e r i f y i n g v a r i o u s v a l u e s o f t

(> 3) a n d o f c o u r s e u p t o 9

l e t t = 7 a n d h = 3 s a t i s f y t h e e q u a t i o n :

h (100 - h^{2}) = 7 (7^{2} - 10)

h^{3} - 100 h + 273 = 0

h = 3

H e n c e t w o r e q u i r e d n u m b e r s

a r e : 370 & 371

^{∙} F o r u > 1 t h i s a p p r o a c h i s

d i f f i c u l t .

Commented by I want to learn more last updated on 31/Aug/20

Thanks sir, i appreciate .