Puzzle for August 10, 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.
* AB, DE, and EF are 2-digit numbers (not A×B, D×E, or E×F).
Scratchpad
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Hint #1
eq.4 may be written as: F – A + E = C + E Subtract E from each side of the above equation: F – A + E – E = C + E – E which becomes eq.4a) F – A = C
Hint #2
In eq.2, replace C with F – A (from eq.4a): D = A + F – A which makes D = F
Hint #3
In eq.3, replace F with D: C + D = A + D Subtract D from both sides of the above equation: C + D – D = A + D – D which makes C = A
Hint #4
In eq.2, substitute A for C: D = A + A which makes D = 2×A and also makes F = D = 2×A
Hint #5
Substitute A for C in eq.5: B ÷ A = F ÷ A Multiply both sides of the equation above by A: (B ÷ A) × A = (F ÷ A) × A which makes B = F and also makes B = F = D = 2×A
Hint #6
eq.6 may be written as: 10×D + E = 10×A + B + 10×E + F Substitute (2×A) for D, B, and F in the above equation: 10×(2×A) + E = 10×A + (2×A) + 10×E + (2×A) which becomes 20×A + E = 14×A + 10×E Subtract E and 14×A from both sides: 20×A + E – E – 14×A = 14×A + 10×E – E – 14×A which simplifies to 6×A = 9×E Divide both sides by 9: 6×A ÷ 9 = 9×E ÷ 9 which makes ⅔×A = E
Solution
Substitute 2×A for B and D and F, A for C, and ⅔×A for E in eq.1: A + 2×A + A + 2×A + ⅔×A + 2×A = 26 which simplifies to 8⅔×A = 26 Divide both sides of the above equation by 8⅔: 8⅔×A ÷ 8⅔ = 26 ÷ 8⅔ which means A = 3 making B = D = F = 2×A = 2 × 3 = 6 C = A = 3 E = ⅔×A = ⅔ × 3 = 2 and ABCDEF = 363626