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Question Number 200836 by Rasheed.Sindhi last updated on 24/Nov/23

L e t \overset{―}{a b c} + \overset{―}{b c a} + \overset{―}{c a b} = \overset{―}{d e f g}

w h e r e a, b, \dots, g a r e d e c i m a l d i g i t s

(m a y b e e q u a l t o 0)

S h o w t h a t

(i) \overset{―}{d g} = a + b + c

(i i) e = f = d + g

Commented by LaChicaFrancesa last updated on 24/Nov/23

Hola, soy nueva en la app y quería consultar cómo se hace para que las 3 letras lleven encima la barra. Qué símbolo se usa?

Commented by AST last updated on 24/Nov/23

D o y o u m e a n \overset{―}{a b c} + \overset{―}{b c a} + \overset{―}{c a b} ?

Commented by Rasheed.Sindhi last updated on 24/Nov/23

Y e s s i r, e x a c t l y!

Commented by AST last updated on 24/Nov/23

\overset{―}{d g} = a + b + c i s n o t t r u e f o r a + b + c = 19

s i n c e \overset{―}{d g} = 29, F o r e x a m p l e a = 9, b = 8, c = 2

Commented by Rasheed.Sindhi last updated on 25/Nov/23

(i) U s e f o r u m k e y b o a r d

(i i) S e l e c t t h e^{'} {w o r d}^{'} w h i c h y o u

w a n t m a k e u n d e r b a r .

(i i i) T a p o n \overset{}{}

(i v) C l i c k o n^{'} -^{'} (m i n u s s i g n)

Answered by AST last updated on 24/Nov/23

p = \overset{―}{a b c} + \overset{―}{b c a} + \overset{―}{c a b} = 111 (a + b + c) ⩽ 2997 \Rightarrow d ⩽ 2

\overset{―}{d e f g} = 1000 d + 100 e + 10 f + g \equiv g \overset{10}{\equiv} a + b + c

d = 0 \Rightarrow p ⩽ 999 \Rightarrow a + b + c ⩽ 9

\Rightarrow a + b + c = g = 10 (0) + g = 10 d + g = \overset{―}{d g}

d = 1 \Rightarrow 10 ⩽ a + b + c ⩽ 18 \Rightarrow a + b + c = 10 (1) + q \overset{10}{\equiv} q \overset{10}{\equiv} g

1 ⩽ q ⩽ 8 \Rightarrow q = g \Rightarrow a + b + c = 10 (1) + g = 10 d + g = \overset{―}{d g}

d = 2 \Rightarrow 19 ⩽ a + b + c ⩽ 27,

F o r 20 ⩽ a + b + c ⩽ 27; s i m i l a r a r g u m e n t a p p l i e s

S u p p o s e a + b + c = 19 \Rightarrow \overset{―}{d e f g} = 2109 \Rightarrow \overset{―}{d g} = 29 \neq 19

T h i s a l s o c o n t r a d i c t s (i i) .

Commented by Rasheed.Sindhi last updated on 24/Nov/23

V . nice analysis!

T han k s sir!

T h e s t a t e m e n t i s n o t t r u e i n a

f e w c a s e s, o f c o u r s e . H o w e v e r

y o u r c o u n t e r e x a m p l e m a y b e

j u s t i f i e d i f w e b e a l l o w e d d e c i m a l

d i g i t s g r e a t e r t h a n 9 a l s o .

L e t X i s d i g i t h a v i n g v a l u e 10

982 + 829 + 298 = 1 XX 9

= 1000 + 100 (10) + 10 (10) + 9 = 2109

A n y w a y {y o u}^{'} r e v e r y r i g h t b e c a u s e

d e c i m a l d i g i t m a y n o t e x c e e d

t h a n 9 .