Puzzle for October 10, 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 A with C + D (from eq.2): C + D + D = C + E which becomes C + 2×D = C + E Subtract C from each side of the equation above: C + 2×D - C = C + E - C which makes 2×D = E
Hint #2
In eq.3, replace E with 2×D: F = D + 2×D which makes F = 3×D
Hint #3
eq.6 may be written as: F = (C + D + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × F = 3 × (C + D + E) ÷ 3 which becomes eq.6a) 3×F = C + D + E
Hint #4
In eq.6a, substitute 3×D for F, and 2×D for E: 3×(3×D) = C + D + 2×D which becomes 9×D = C + 3×D Subtract 3×D from both sides of the equation above: 9×D - 3×D = C + 3×D - 3×D which makes eq.6a) 6×D = C
Hint #5
Substitute 6×D for C in eq.2: A = 6×D + D which makes A = 7×D
Hint #6
Substitute 7×D for A, and 2×D for E in eq.4: B + D = 7×D + 2×D which becomes B + D = 9×D Subtract D from both sides of the above equation: B + D - D = 9×D - D which makes B = 8×D
Solution
Substitute 7×D for A, 8×D for B, 6×D for C, 2×D for E, and 3×D for F in eq.1: 7×D + 8×D + 6×D + D + 2×D + 3×D = 27 which simplifies to 27×D = 27 Divide both sides of the above equation by 27: 27×D ÷ 27 = 27 ÷ 27 which means D = 1 making A = 7×D = 7×1 = 7 B = 8×D = 8×1 = 8 C = 6×D = 6×1 = 6 E = 2×D = 2×1 = 2 F = 3×D = 3×1 = 3 and ABCDEF = 786123