Puzzle for August 26, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
In eq.5, replace E with C – A (from eq.4): C + C – A = B – A which becomes 2×C – A = B – A Add A to both sides of the above equation: 2×C – A + A = B – A + A which makes eq.5a) 2×C = B
Hint #2
In eq.6, replace B with 2×C (from eq.5a): 2×C ÷ C = E which makes 2 = E
Hint #3
In eq.4, substitute 2 for E: 2 = C – A In the above equation, add A to both sides, and subtract 2 from both sides: 2 + A – 2 = C – A + A – 2 which means eq.4a) A = C – 2
Hint #4
Substitute 2×C for B, and 2 for E in eq.2: 2×C = 2 + F Subtract 2 from both sides of the above equation: 2×C – 2 = 2 + F – 2 which makes eq.2a) 2×C – 2 = F
Hint #5
Substitute 2×C – 2 for F (from eq.2a), and C – 2 for A (from eq.4a) in eq.3: D + 2×C – 2 = C – 2 + C which becomes D + 2×C – 2 = 2×C – 2 Add 2 to both sides, and subtract 2×C from both sides: D + 2×C – 2 + 2 – 2×C = 2×C – 2 + 2 – 2×C which means D = 0
Solution
Substitute C – 2 for A (from eq.4a), 2×C for B, 0 for D, 2 for E, and 2×C – 2 for F (from eq.2a) in eq.1: C – 2 + 2×C + C + 0 + 2 + 2×C – 2 = 16 which simplifies to 6×C – 2 = 16 Add 2 to both sides of the above equation: 6×C – 2 + 2 = 16 + 2 which makes 6×C = 18 Divide both sides by 6: 6×C ÷ 6 = 18 ÷ 6 which means C = 3 making A = C – 2 = 3 – 2 = 1 (from eq.4a) B = 2×C = 2×3 = 6 F = 2×C – 2 = 2×3 – 2 = 6 – 2 = 4 (from eq.2a) and ABCDEF = 163024