Puzzle for March 13, 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.4, replace B with C + E (from eq.2): D + E = C + E - D In the above equation, subtract E from both sides, and add D to both sides: D + E - E + D = C + E - D - E + D which simplifies to 2×D = C
Hint #2
In eq.5, replace C with 2×D: E - D = 2×D - E Add D and E to both sides of the above equation: E - D + D + E = 2×D - E + D + E which makes 2×E = 3×D Divide both sides by 2: 2×E ÷ 2 = 3×D ÷ 2 which makes E = 1½×D
Hint #3
In eq.2, substitute 2×D for C, and 1½×D for E: B = 2×D + 1½×D which makes B = 3½×D
Hint #4
In eq.3, substitute 2×D for C, and 1½×D for E: F = 2×D + D + 1½×D which makes F = 4½×D
Hint #5
Substitute 2×D for C in eq.6: A - D = 2×D + D which becomes A - D = 3×D Add D to both sides of the above equation: A - D + D = 3×D + D which makes A = 4×D
Solution
Substitute 4×D for A, 3½×D for B, 2×D for C, 1½×D for E, and 4½×D for F in eq.1: 4×D + 3½×D + 2×D + D + 1½×D + 4½×D = 33 which simplifies to 16½×D = 33 Divide both sides of the above equation by 16½: 16½×D ÷ 16½ = 33 ÷ 16½ which means D = 2 making A = 4×D = 4 × 2 = 8 B = 3½×D = 3½ × 2 = 7 C = 2×D = 2 × 2 = 4 E = 1½×D = 1½ × 2 = 3 F = 4½×D = 4½ × 2 = 9 and ABCDEF = 874239