Puzzle for June 8, 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.4, replace A with C + D (from eq.3): D + F = C + D + C which becomes D + F = 2×C + D Subtract D from both sides of the above equation: D + F – D = 2×C + D – D which makes F = 2×C
Hint #2
In eq.2, replace F with 2×C: E = C + 2×C which makes E = 3×C
Hint #3
eq.6 may be written as: A – E + A = E + F – A which becomes 2×A – E = E + F – A Add E and A to both side of the above equation: 2×A – E + E + A = E + F – A + E + A which becomes eq.6a) 3×A = 2×E + F
Hint #4
Substitute (3×C) for E, and 2×C for F in eq.6a: 3×A = 2×(3×C) + 2×C which becomes 3×A = 6×C + 2×C which makes 3×A = 8×C Divide both sides of the above equation by 3: 3×A ÷ 3 = 8×C ÷ 3 which makes A = 2⅔×C
Hint #5
Substitute 2⅔×C for A in eq.3: C + D = 2⅔×C Subtract C from each side of the above equation: C + D – C = 2⅔×C – C which makes D = 1⅔×C
Hint #6
Substitute 1⅔×C for D, 2⅔×C for A, and 3×C for E in eq.5: B + C + 1⅔×C = 2⅔×C – C + 3×C which becomes B + 2⅔×C = 4⅔×C Subtract 2⅔×C from both sides of the equation above: B + 2⅔×C – 2⅔×C = 4⅔×C – 2⅔×C which becomes B = 2×C
Solution
Substitute 2⅔×C for A, 2×C for B and F, 1⅔×C for D, and 3×C for E in eq.1: 2⅔×C + 2×C + C + 1⅔×C + 3×C + 2×C = 37 which simplifies to 12⅓×C = 37 Divide both sides of the equation above by 12⅓: 12⅓×C ÷ 12⅓ = 37 ÷ 12⅓ which means C = 3 making A = 2⅔×C = 2⅔ × 3 = 8 B = F = 2×C = 2 × 3 = 6 D = 1⅔×C = 1⅔ × 3 = 5 E = 3×C = 3 × 3 = 9 and ABCDEF = 863596