Puzzle for July 17, 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 E with B + D (from eq.2): D + B + D = A + B which becomes 2×D + B = A + B Subtract B from each side of the equation above: 2×D + B - B = A + B - B which makes eq.5a) 2×D = A
Hint #2
In eq.6, replace A with 2×D: C + D = 2×D - C In the above equation, subtract D from both sides, and add C to both sides: C + D - D + C = 2×D - C - D + C which makes 2×C = D
Hint #3
In eq.5a, substitute (2×C) for D: 2×(2×C) = A which makes 4×C = A
Hint #4
Substitute 4×C for A into eq.3: B = 4×C + C which makes B = 5×C
Hint #5
Substitute 5×C for B, and 2×C for D in eq.2: E = 5×C + 2×C which makes E = 7×C
Hint #6
Substitute 5×C for B in eq.4: F = 5×C + C which makes F = 6×C
Solution
Substitute 4×C for A, 5×C for B, 2×C for D, 7×C for E, and 6×C for F in eq.1: 4×C + 5×C + C + 2×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 = 4×C = 4 × 1 = 4 B = 5×C = 5 × 1 = 5 D = 2×C = 2 × 1 = 2 E = 7×C = 7 × 1 = 7 F = 6×C = 6 × 1 = 6 and ABCDEF = 451276