Puzzle for April 6, 2023 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit positive integer.
* AB and DE are 2-digit numbers (not A×B or D×E).
Scratchpad
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Hint #1
Add B and E to both sides of eq.2: F - B + B + E = A - E + B + E which becomes eq.2a) F + E = A + B
Hint #2
In eq.1, replace A + B with F + E (from eq.2a): D + F = F + E Subtract F from each side of the equation above: D + F - F = F + E - F which becomes D = E
Hint #3
In eq.4, replace E with D: eq.4a) A + D = B + C
Hint #4
Subtract the left and right sides of eq.3 from the left and right sides of eq.4a, respectively: A + D - (A + C) = B + C - (B + D) which becomes A + D - A - C = B + C - B - D which becomes D - C = C - D Add C and D to both sides of the above equation: D - C + C + D = C - D + C + D which makes 2×D = 2×C Divide both sides by 2: 2×D ÷ 2 = 2×C ÷ 2 which makes D = C
Hint #5
In eq.4a, substitute C for D: A + C = B + C Subtract C from each side of the above equation: A + C - C = B + C - C which makes A = B
Hint #6
eq.5 may be written as: 10×A + B + C = 10×D + E - A In the above equation, subtract C from both sides, and add A to both sides: 10×A + B + C - C + A = 10×D + E - A - C + A which makes eq.5a) 11×A + B = 10×D + E - C
Hint #7
In eq.5a, substitute A for B, and D for E and C: 11×A + A = 10×D + D - D which makes 12×A = 10×D Divide both sides of the above equation by 10: 12×A ÷ 10 = 10×D ÷ 10 which makes 1⅕×A = D and also makes eq.5b) 1⅕×A = C = D = E
Hint #8
Substitute A for B, and C for D and E in eq.6: A × C = C + (C × F) which may be written as A × C = C × (1 + F) Divide both sides of the above equation by C: (A × C) ÷ C = (C × (1 + F)) ÷ C which becomes eq.6a) A = 1 + F
Hint #9
Substitute 1⅕×A for E (from eq.5b), and A for B in eq.2a: F + 1⅕×A = A + A which becomes F + 1⅕×A = 2×A Subtract 1⅕×A from each side of the above equation: F + 1⅕×A - 1⅕×A = 2×A - 1⅕×A which makes F = ⅘×A
Solution
Substitute ⅘×A for F in eq.6a: A = 1 + ⅘×A Subtract ⅘×A from each side of the equation above: A - ⅘×A = 1 + ⅘×A - ⅘×A which makes ⅕×A = 1 Multiply both sides by 5: 5 × ⅕×A = 5 × 1 which makes A = 5 making B = A = 5 C = D = E = 1⅕×A = 1⅕ × 5 = 6 (from eq.5b) F = ⅘×A = ⅘ × 5 = 4 and ABCDEF = 556664