Puzzle for May 13, 2019 ( )
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
Add C to both sides of eq.2: C + F + C = D - C + C which becomes 2×C + F = D In eq.6, replace D with 2×C + F: A + F = 2×C + F Subtract F from both sides of the above equation: A + F - F = 2×C + F - F which makes A = 2×C
Hint #2
In eq.3, replace A with 2×C: B = 2×C + C + F which becomes eq.3a) B = 3×C + F In eq.4, substitute 3×C + F for B: E + F = 3×C + F + C Subtract F from both sides: E + F - F = 3×C + F + C - F which makes E = 4×C
Hint #3
Substitute 4×C for E, and 2×C for A in eq.5: D - 4×C = 2×C + C - D Add 4×C + D to each side of the equation above: D - 4×C + 4×C + D = 2×C + C - D + 4×C + D which simplifies to 2×D = 7×C Divide both sides by 2: 2×D ÷ 2 = 7×C ÷ 2 which means D = 3½×C
Hint #4
Substitute 2×C for A, and 3½×C for D in eq.6: 2×C + F = 3½×C Subtract 2×C from both sides: 2×C + F - 2×C = 3½×C - 2×C which makes F = 1½×C
Hint #5
Substitute 1½×C for F in eq.3a: B = 3×C + 1½×C which makes B = 4½×C
Solution
Substitute 2×C for A, 4½×C for B, 3½×C for D, and 4×C for E, and 1½×C for F in eq.1: 2×C + 4½×C + C + 3½×C + 4×C + 1½×C = 33 which simplifies to 16½×C = 33 Divide both sides of the above equation by 16½: 16½×C ÷ 16½ = 33 ÷ 16½ which means C = 2 making A = 2×C = 2 × 2 = 4 B = 4½×C = 4½ × 2 = 9 D = 3½×C = 3½ × 2 = 7 E = 4×C = 4 × 2 = 8 F = 1½×C = 1½ × 2 = 3 and ABCDEF = 492783