Puzzle for May 1, 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 non-negative integer.
Scratchpad
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Hint #1
In eq.5, replace B with D + F (from eq.2): F - D = D + F - F which becomes F - D = D Add D to both sides of the above equation: F - D + D = D + D which makes 2×D = F
Hint #2
In eq.2, replace F with 2×D: B = D + 2×D which makes B = 3×D
Hint #3
In eq.3, replace F with 2×D: eq.3a) C = E + 2×D
Hint #4
In eq.6, substitute 3×D for B, and E + 2×D for C (from eq.3a): 3×D + E + 2×D = A + E - 3×D which becomes 5×D + E = A + E - 3×D In the above equation, subtract E from both sides, and add 3×D to both sides: 5×D + E - E + 3×D = A + E - 3×D - E + 3×D which simplifies to 8×D = A
Hint #5
Substitute 8×D for A in eq.4: 8×D = D + E Subtract D from each side of the equation above: 8×D - D = D + E - D which makes 7×D = E
Hint #6
Substitute 7×D for E in eq.3a: C = 7×D + 2×D which makes C = 9×D
Solution
Substitute 8×D for A, 3×D for B, 9×D for C, 7×D for E, and 2×D for F in eq.1: 8×D + 3×D + 9×D + D + 7×D + 2×D = 30 which simplifies to 30×D = 30 Divide both sides of the above equation by 30: 30×D ÷ 30 = 30 ÷ 30 which means D = 1 making A = 8×D = 8 × D = 8 B = 3×D = 3 × D = 3 C = 9×D = 9 × D = 9 E = 7×D = 7 × D = 7 F = 2×D = 2 × D = 2 and ABCDEF = 839172