Let abc ^(−) + bca ^(−) + cab ^(−) = defg ^(−) where a,b,...,g are decimal digits (may be equal to 0) Show that (i) dg ^(−) =a+b+c (ii) e=f=d+g


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

$L e t \overset{―}{a b c} + \overset{―}{b c a} + \overset{―}{c a b} = \overset{―}{d e f g}$ $w h e r e a, b, . . ., g a r e d e c i m a l d i g i t s$ $(m a y b e e q u a l t o 0)$ $S h o w t h a t$ $(i) \overset{―}{d g} = a + b + c$ $(i i) e = f = d + g$
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 deleteduser1 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 deleteduser1 last updated on 24/Nov/23

$\overset{―}{d g} = a + b + c i s n o t t r u e f o r a + b + c = 19$ $s i n c e \overset{―}{d g} = 29, F o r e x a m p l e a = 9, b = 8, c = 2$
	Answered by deleteduser1 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

	$V . nice analysis!$ $T han k s sir!$ $T h e s t a t e m e n t i s n o t t r u e i n a$ $f e w c a s e s, o f c o u r s e . H o w e v e r$ $y o u r c o u n t e r e x a m p l e m a y b e$ $j u s t i f i e d i f w e b e a l l o w e d d e c i m a l$ $d i g i t s g r e a t e r t h a n 9 a l s o .$ $L e t X i s d i g i t h a v i n g v a l u e 10$ $982 + 829 + 298 = 1 XX 9$ $= 1000 + 100 (10) + 10 (10) + 9 = 2109$ $A n y w a y {y o u}^{'} r e v e r y r i g h t b e c a u s e$ $d e c i m a l d i g i t m a y n o t e x c e e d$ $t h a n 9 .$
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