Puzzle for April 23, 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.
* BC is a 2-digit number (not B×C). ABC and DEF are 3-digit numbers (not A×B×C or D×E×F).
Scratchpad
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Hint #1
eq.6 may be written as: 100×A + 10×B + C + F = 100×D + 10×E + F + 10×B + C Subtract 10×B, C, and F from each side of the above equation: 100×A + 10×B + C + F - 10×B - C - F = 100×D + 10×E + F + 10×B + C - 10×B - C - F which simplifies to 100×A = 100×D + 10×E Divide both sides by 10: 100×A ÷ 10 = (100×D + 10×E) ÷ 10 which becomes 10×A = 10×D + E Subtract 10×D from both sides: 10×A - 10×D = 10×D + E - 10×D which becomes eq.6a) 10×A - 10×D = E
Hint #2
In eq.3, substitute (10×A - 10×D) for E (from eq.6a): D - (10×A - 10×D) = A which may be written as D - 10×A + 10×D = A which becomes 11×D - 10×A = A Add 10×A to both sides of the above equation: 11×D - 10×A + 10×A = A + 10×A which makes 11×D = 11×A Divide both sides by 11: 11×D ÷ 11 = 11×A ÷ 11 which means D = A
Hint #3
In eq.5, substitute D for A: B = D + D + (D ÷ D) which becomes eq.5a) B = 2×D + 1 (implies D ≠ 0)
Hint #4
In eq.3, substitute D for A: D - E = D Subtract D from both sides of the equation above: D - E - D = D - D which makes -E = 0 which means E = 0
Hint #5
Substitute 0 for E (in eq.3): C + 0 = B which makes C = B which also means eq.3a) C = B = 2×D + 1 (from eq.5a)
Hint #6
In eq.4, substitute 0 for E, and add F to both sides: F - 0 + F = D + 0 - F + F which simplifies to 2×F = D Divide both sides by 2: 2×F ÷ 2 = D ÷ 2 which means F = ½×D
Solution
Substitute D for A, 2×D + 1 for B and C (from eq.3a), 0 for E, and ½×D for F in eq.1: D + 2×D + 1 + 2×D + 1 + D + 0 + ½×D = 28 which becomes 6½×D + 2 = 28 Subtract 2 from each side of the equation above: 6½×D + 2 - 2 = 28 - 2 which becomes 6½×D = 26 Divide both sides by 6½: 6½×D ÷ 6½ = 26 ÷ 6½ which means D = 4 making A = D = 4 B = C = 2×D + 1 = 2×4 + 1 = 8 + 1 = 9 (from eq.5a) F = ½×D = ½ × 4 = 2 and ABCDEF = 499402