Puzzle for August 12, 2021 ( )
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 and EF are 2-digit numbers (not B×C or E×F).
Scratchpad
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Hint #1
Multiply both sides of eq.5 by D: D × E = D × (B + E + F) ÷ D which becomes eq.5a) D × E = B + E + F Multiply both sides of eq.6 by C: C × D = C × (B + D + E + F) ÷ C which becomes C × D = B + D + E + F which may be written as eq.6a) C × D = D + B + E + F
Hint #2
In eq.6a, substitute (D × E) for B + E + F (from eq.5a): C × D = D + (D × E) which may be written as C × D = D × (1 + E) Since D ≠ 0 (from eq.5), divide both sides of the above equation by D: (C × D) ÷ D = (D × (1 + E)) ÷ D which makes eq.6b) C = 1 + E
Hint #3
In eq.1, replace C with 1 + E (from eq.6b): 1 + E = A + E Subtract E from each side of the above equation: 1 + E – E = A + E – E which makes 1 = A
Hint #4
In eq.4, replace A with 1: 1 = F ÷ B Multiply both sides of the above equation by B: 1 × B = (F ÷ B) × B which makes B = F
Hint #5
eq.3 may be written as: 10×B + C – (10×E + F) = A which is the same as 10×B + C – 10×E – F = A In the equation above, substitute 1 + E for C (from eq.6b), B for F, and 1 for A: 10×B + 1 + E – 10×E – B = 1 which becomes 9×B + 1 – 9×E = 1 Subtract 1 from both sides, and add 9×E to both sides: 9×B + 1 – 9×E – 1 + 9×E = 1 – 1 + 9×E which makes 9×B = 9×E Divide both sides by 9: 9×B ÷ 9 = 9×E ÷ 9 which makes B = E
Hint #6
Substitute B for both E and F in eq.5: B = (B + B + B) ÷ D which becomes B = (3×B) ÷ D Multiply both sides of the above equation by D: D × B = D × ((3×B) ÷ D) which makes D × B = 3×B Since B ≠ 0 (from eq.4), divide both sides by B: (D × B) ÷ B = (3×B) ÷ B which makes D = 3
Solution
Substitute 1 for A, and 3 for D in eq.2: F – 1 – 3 = 1 + 3 which becomes F – 4 = 4 Add 4 to both sides of the above equation: F – 4 + 4 = 4 + 4 which makes F = 8 and makes B = E = F = 8 C = 1 + E = 1 + 8 = 9 (from eq.6b) and ABCDEF = 189388