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Question Number 215659 by Ari last updated on 13/Jan/25

	Answered by Red1ight last updated on 13/Jan/25

	$2 (100 x + 10 y + z) = (x + 10 y) + (x + 10 z)$ $+ (y + 10 x) + (y + 10 z) + (z + 10 x) + (z + 10 y)$ $⋮$ $200 x + 20 y + 2 z = 22 x + 22 y + 22 z$ $- 178 x + 2 y + 20 z = 0$ $x, y, z \in W, x, y, z < 10$ $L e t z, y = 9$ $- 178 x + 198 = 0$ $0 < x < 2$ $∴ x = 1$ $z = 8$ $y = 9$ $The number = 198$
	Commented by mr W last updated on 13/Jan/25

	$r i g h t!$ $y = 89 x - 10 z$ $0 < 89 x - 10 z ⩽ 9$ $10 ⩽ 10 z < 89 x ⩽ 9 + 10 z ⩽ 99$ $\Rightarrow 1 ⩽ x ⩽ 1 \Rightarrow x = 1$ $0 < 89 - 10 z ⩽ 9$ $10 z < 89 ⩽ 9 + 10 z$ $\Rightarrow 8 ⩽ z ⩽ 8 \Rightarrow z = 8$ $\Rightarrow y = 89 - 80 = 9$ $t h e o n l y o n e n u m b e r i s 198 .$
	Answered by Rasheed.Sindhi last updated on 14/Jan/25
	$=−=−=−=−=−=−=−=−=−=−=−=−=−= 2(abc^(−) )=ab^(−) +ba^(−) +bc^(−) +cb^(−) +ac^(−) +ca^(−) 2(abc^(−) )=11(a+b)+11(b+c)+11(c+a) =11×2(a+b+c) determinant (((abc^(−) =11(a+b+c)...A))) A⇒100a+10b+c=11a+11b+11c ⇒89a−b−10c=0....(i) Again, A⇒11∣ abc^(−) ⇒ { ((a+c−b=0...(ii) )),(( or)),((a+c−b=11...(iii))) :} (i) & (ii): { ((89a−b−10c=0)),((a+c−b=0)) :} ⇒88a−11c=0⇒8a=c⇒a=1,c=8 a+c−b=0⇒b=a+c=1+8=9 abc^(−) =198 ✓ (i) & (iii): { ((89a−b−10c=0)),((a+c−b=11)) :} 88a−11c=−11⇒c=8a+1⇒a=1,c=9 ⇒a+c−b=11 ⇒b=a+c−11=1+9−11=−1 Rejected$
	$= - = - = - = - = - = - = - = - = - = - = - = - = - =$ $2 (\overset{―}{a b c}) = \overset{―}{a b} + \overset{―}{b a} + \overset{―}{b c} + \overset{―}{c b} + \overset{―}{a c} + \overset{―}{c a}$ $2 (\overset{―}{a b c}) = 11 (a + b) + 11 (b + c) + 11 (c + a)$ $= 11 \times 2 (a + b + c)$ $\begin{matrix} \overset{―}{a b c} = 11 (a + b + c) . . . A \end{matrix}$ $A \Rightarrow 100 a + 10 b + c = 11 a + 11 b + 11 c$ $\Rightarrow 89 a - b - 10 c = 0 . . . . (i)$ $A g a i n,$ $A \Rightarrow 11 ∣ \overset{―}{a b c} \Rightarrow {\begin{cases} a + c - b = 0 . . . (ii) \\ or \\ a + c - b = 11 . . . (iii) \end{cases}$ $(i) & (ii) : {\begin{cases} 89 a - b - 10 c = 0 \\ a + c - b = 0 \end{cases}$ $\Rightarrow 88 a - 11 c = 0 \Rightarrow 8 a = c \Rightarrow a = 1, c = 8$ $a + c - b = 0 \Rightarrow b = a + c = 1 + 8 = 9$ $\overset{―}{a b c} = 198 ✓$ $(i) & (iii) : {\begin{cases} 89 a - b - 10 c = 0 \\ a + c - b = 11 \end{cases}$ $88 a - 11 c = - 11 \Rightarrow c = 8 a + 1 \Rightarrow a = 1, c = 9$ $\Rightarrow a + c - b = 11$ $\Rightarrow b = a + c - 11 = 1 + 9 - 11 = - 1$ $R e j e c t e d$
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