Puzzle for May 15, 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.
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Hint #1
In eq.6, replace E with A + B (from eq.3): B + D = A + A + B which becomes B + D = 2×A + B Subtract B from each side of the equation above: B + D - B = 2×A + B - B which makes eq.6a) D = 2×A
Hint #2
In eq.5, replace D with 2×A: 2×A - C = A + C In the above equation, add C to both sides, and subtract A from both sides: 2×A - C + C - A = A + C + C - A which makes A = 2×C
Hint #3
In eq.6a, substitute (2×C) for A: D = 2×(2×C) which makes D = 4×C
Hint #4
Substitute 2×C for A, and 4×C for D in eq.2: F = 2×C + 4×C which makes F = 6×C
Hint #5
Substitute 4×C for D in eq.4: B = C + 4×C which makes B = 5×C
Hint #6
Substitute 2×C for A, and 5×C for B in eq.3: E = 2×C + 5×C which makes E = 7×C
Solution
Substitute 2×C for A, 5×C for B, 4×C for D, 7×C for E, and 6×C for F in eq.1: 2×C + 5×C + C + 4×C + 7×C + 6×C = 25 which simplifies to 25×C = 25 Divide both sides of the above equation by 25: 25×C ÷ 25 = 25 ÷ 25 which means C = 1 making A = 2×C = 2 × 1 = 2 B = 5×C = 5 × 1 = 5 D = 4×C = 4 × 1 = 4 E = 7×C = 7 × 1 = 7 F = 6×C = 6 × 1 = 6 and ABCDEF = 251476