Puzzle for August 9, 2021 ( )
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 A + C (from eq.3): C + A + C = A + F which becomes 2×C + A = A + F Subtract A from each side of the equation above: 2×C + A – A = A + F – A which makes 2×C = F
Hint #2
In eq.4, replace F with 2×C: A = C + 2×C which makes A = 3×C
Hint #3
In eq.3, substitute 3×C for A: E = 3×C + C which makes E = 4×C
Hint #4
Substitute 4×C for E, and 3×C for A in eq.6: 4×C – B = 3×C + B In the above equation, add B to both sides, and subtract 3×C from both sides: 4×C – B + B – 3×C = 3×C + B + B – 3×C which makes C = 2×B Divide both sides by 2: C ÷ 2 = 2×B ÷ 2 which makes ½×C = B
Hint #5
Substitute ½×C for B in eq.2: D = ½×C + C which makes D = 1½×C
Solution
Substitute 3×C for A, ½×C for B, 1½×C for D, 4×C for E, and 2×C for F in eq.1: 3×C + ½×C + C + 1½×C + 4×C + 2×C = 24 which simplifies to 12×C = 24 Divide both sides of the above equation by 12: 12×C ÷ 12 = 24 ÷ 12 which means C = 2 making A = 3×C = 3 × 2 = 6 B = ½×C = ½ × 2 = 1 D = 1½×C = 1½ × 2 = 3 E = 4×C = 4 × 2 = 8 F = 2×C = 2 × 2 = 4 and ABCDEF = 612384