Lê Quốc Trần Anh Coordinator
03/01/2018 at 16:56-
ABC + DBBB = 2011
=> 100A + 10B + C + 1000D + 100B + 10B + B = 2011
=> 100A + 1121B + 1000D + C = 2011
Because A,B,C,D are digits
=> \(\left\{{}\begin{matrix}9\ge A,D>0\\9\ge B,C\ge0\end{matrix}\right.\)
If B = 2 => 100A + 1121B + 1000D + C = 100A + 2242 + 1000D + C > 2011 [unsatisfactory]
=> B = 0 or 1
If D = 2 => 100A + 1121B + 1000D + C = 100A + 1121B + 2000 + C = 2011
<=> 100A + 1121B + C = 11 but 100A \(\ge100\) [unsatisfactory]
=> D = 0 or 1
** B = 0, D = 0
=> 100A + 1121B + 1000D + C = 100A + 0 + 0 + C = 2011
But 101 \(\le\) 100A + C \(\le909\)
<=> unsatisfactory
** B = 0; D = 1
=> 100A + 1121B + 1000D + C = 100A + 0 + 1000 + C = 2011
<=> 100A + C = 1011 [the same this case]
<=> unsatisfactory
** B = 1; D = 0
=> 100A + 1121B + 1000D + C = 100A + 1121 + 0 + C = 2011
=> 100A + C = 890
But C < 10 => A = 8 <=> 800 + C = 890 <=> C = 90 > 10 [unsatisfactory]
** B=1; D = 1
=> 100A + 1121B + 1000D + C = 100A + 1121 + 1000 + C = 2011
=> 100A + C = - 11 [ unsatisfactory ]
So there are no numbers are satisfactory
So A + B + C + D \(\in\phi\)
Lê Quốc Trần Anh selected this answer. -
FA Liên Quân Garena 08/01/2018 at 21:50BC + DBBB = 2011
=> 100A + 10B + C + 1000D + 100B + 10B + B = 2011
=> 100A + 1121B + 1000D + C = 2011
Because A,B,C,D are digits
=> {9≥A,D>09≥B,C≥0
If B = 2 => 100A + 1121B + 1000D + C = 100A + 2242 + 1000D + C > 2011 [unsatisfactory]
=> B = 0 or 1
If D = 2 => 100A + 1121B + 1000D + C = 100A + 1121B + 2000 + C = 2011
<=> 100A + 1121B + C = 11 but 100A ≥100
[unsatisfactory]
=> D = 0 or 1
** B = 0, D = 0
=> 100A + 1121B + 1000D + C = 100A + 0 + 0 + C = 2011
But 101 ≤
100A + C ≤909
<=> unsatisfactory
** B = 0; D = 1
=> 100A + 1121B + 1000D + C = 100A + 0 + 1000 + C = 2011
<=> 100A + C = 1011 [the same this case]
<=> unsatisfactory
** B = 1; D = 0
=> 100A + 1121B + 1000D + C = 100A + 1121 + 0 + C = 2011
=> 100A + C = 890
But C < 10 => A = 8 <=> 800 + C = 890 <=> C = 90 > 10 [unsatisfactory]
** B=1; D = 1
=> 100A + 1121B + 1000D + C = 100A + 1121 + 1000 + C = 2011
=> 100A + C = - 11 [ unsatisfactory ]
So there are no numbers are satisfactory
So A + B + C + D ∈ϕ