Puzzle for June 10, 2023 ( )
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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
Multiply both sides of eq.5 by F: F × (A + D) = F × (B ÷ F) which becomes eq.5a) F × (A + D) = B
Hint #2
Add A and D to both sides of eq.6: B - A - D + A + D = A + D + A + D which becomes B = 2×A + 2×D which may be written as eq.6a) B = 2×(A + D)
Hint #3
In eq.5a, substitute 2×(A + D) for B (from eq.6a): eq.5b) F × (A + D) = 2×(A + D) Since A and D are non-negative integers, and since D ≠ 0 (from eq.4), then: A + D ≠ 0 Divide both sides of eq.5b by (A + D): F × (A + D) ÷ (A + D) = 2×(A + D) ÷ (A + D) which simplifies to F = 2
Hint #4
In eq.4, replace F with 2: A = 2 ÷ D Multiply both sides of the above equation by D: A × D = (2 ÷ D) × D which becomes eq.4a) A × D = 2
Hint #5
Since A and D are one-digit non-negative integers, eq.4a makes: A = 1 and D = 2 which makes A + D = 3 or: A = 2 and D = 1 which makes A + D = 3 In either of the above cases: eq.4b) A + D = 3
Hint #6
In eq.6a, replace A + D with 3 (from eq.4b): B = 2×(3) which makes B = 6
Hint #7
Subtract the left and right sides of eq.2 from the left and right sides of eq.4b, respectively: A + D - (D - A) = 3 - (A - C) which becomes A + D - D + A = 3 - A + C which becomes 2×A = 3 - A + C In the above equation, subtract 3 from both sides, and add A to both sides: 2×A - 3 + A = 3 - A + C - 3 + A which becomes eq.2a) 3×A - 3 = C
Hint #8
Subtract A from each side of eq.4b: A + D - A = 3 - A which becomes eq.4c) D = 3 - A
Hint #9
Substitute 6 for B, 3×A - 3 for C (from eq.2a), 3 - A for D (from eq.4c), and 2 for F in eq.3: 6 + 3×A - 3 = 3 - A + E + 2 which becomes 3 + 3×A = 5 - A + E In the equation above, subtract 5 from both sides, and add A to both sides: 3 + 3×A - 5 + A = 5 - A + E - 5 + A which becomes -2 + 4×A = E which is the same as eq.3a) 4×A - 2 = E
Hint #10
Substitute 2 for F, 3×A - 3 for C (from eq.2a), and 4×A - 2 for E (from eq.3a) in eq.1: 2 = 3×A - 3 + 4×A - 2 which becomes 2 = 7×A - 5 Add 5 to both sides of the above equation: 2 + 5 = 7×A - 5 + 5 which makes 7 = 7×A Divide both sides by 7: 7 ÷ 7 = 7×A ÷ 7 which makes 1 = A
Solution
Since A = 1, then: C = 3×A - 3 = 3×1 - 3 = 3 - 3 = 0 (from eq.2a) D = 3 - A = 3 - 1 = 2 (from eq.4c) E = 4×A - 2 = 4×1 - 2 = 4 - 2 = 2 (from eq.3a) and ABCDEF = 160222