Puzzle for June 7, 2022 ( )
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.
Scratchpad
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Hint #1
In eq.2, replace C + D with F (from eq.1): F = B – F Add F to both sides of the above equation: F + F = B – F + F which makes 2×F = B
Hint #2
In eq.3, replace B with 2×F: D × F = 2×F + F which becomes D × F = 3×F Divide both sides of the above equation by F: (D × F) ÷ F = 3×F ÷ F which makes D = 3
Hint #3
eq.5 may be written as: D × E = A + C + B + F In the above equation, substitute 3 for D, E for A + C (from eq.4), and 2×F for B: 3 × E = E + 2×F + F which becomes 3×E = E + 3×F Subtract E from each side: 3×E – E = E + 3×F – E which makes 2×E = 3×F Divide both sides by 2: 2×E ÷ 2 = 3×F ÷ 2 which makes E = 1½×F
Hint #4
In eq.1, substitute 3 for D: F = C + 3 Subtract 3 from each side of the equation above: F – 3 = C + 3 – 3 which becomes eq.1a) F – 3 = C
Hint #5
Substitute 2×F for B, and 3 for D in eq.6: 2×F = A + 3 Subtract 3 from each side of the above equation: 2×F – 3 = A + 3 – 3 which becomes eq.6a) 2×F – 3 = A
Hint #6
In eq.4, substitute 1½×F for E, 2×F – 3 for A (from eq.6a), and F – 3 for C (from eq.1a): 1½×F = 2×F – 3 + F – 3 which becomes 1½×F = 3×F – 6 In the above equation, add 6 to both sides, and subtract 1½×F from both sides: 1½×F + 6 – 1½×F = 3×F – 6 + 6 – 1½×F which makes 6 = 1½×F Divide both sides by 1½: 6 ÷ 1½ = 1½×F ÷ 1½ which makes 4 = F
Solution
Since F = 4, then: A = 2×F – 3 = 2×4 – 3 = 8 – 3 = 5 (from eq.6a) B = 2×F = 2×4 = 8 C = F – 3 = 4 – 3 = 1 (from eq.1a) E = 1½×F = 1½ × 4 = 6 and ABCDEF = 581364