Puzzle for November 20, 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 F with B + D (from eq.4): D + B + D = B + E which becomes 2×D + B = B + E Subtract B from each side of the equation above: 2×D + B - B = B + E - B which makes 2×D = E
Hint #2
In eq.3, replace E with 2×D: A = D + 2×D which makes A = 3×D
Hint #3
In eq.6, substitute 3×D for A, and 2×D for E: B = (3×D + 2×D) ÷ D which becomes B = 5×D ÷ D which makes B = 5
Hint #4
Substitute 5 for B in eq.4: eq.4a) F = 5 + D
Hint #5
Substitute 3×D for A, and 5 for B in eq.2: 3×D = 5 + C Subtract 5 from each side of the above equation: 3×D - 5 = 5 + C - 5 which becomes eq.2a) 3×D - 5 = C
Hint #6
Substitute 3×D for A, 5 for B, 3×D - 5 for C (from eq.2a), 2×D for E, and 5 + D for F (from eq.4a) in eq.1: 3×D + 5 + 3×D - 5 + D + 2×D + 5 + D = 35 which simplifies to 10×D + 5 = 35 Subtract 5 from both sides of the above equation: 10×D + 5 - 5 = 35 - 5 which makes 10×D = 30 Divide both sides by 10: 10×D ÷ 10 = 30 ÷ 10 which means D = 3
Solution
Since D = 3, then: A = 3×D = 3×3 = 9 C = 3×D - 5 = 3×3 - 5 = 9 - 5 = 4 (from eq.2a) E = 2×D = 2×3 = 6 F = 5 + D = 5 + 3 = 8 (from eq.4a) and ABCDEF = 954368