Puzzle for June 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.6, replace E with C + D (from eq.4): D + F = B + C + C + D which becomes D + F = B + 2×C + D Subtract D from each side of the equation above: D + F - D = B + 2×C + D - D which becomes eq.6a) F = B + 2×C
Hint #2
In eq.3, replace F with B + 2×C (from eq.6a): B + 2×C = B + D Subtract B from each side of the above equation: B + 2×C - B = B + D - B which makes eq.3a) 2×C = D
Hint #3
In eq.5, substitute 2×C for D: A + C = 2×C - A In the above equation, subtract C from both sides, and add A to both sides: A + C - C + A = 2×C - A - C + A which makes 2×A = C
Hint #4
Substitute 2×A for C in eq.3a: 2×(2×A) = D which makes 4×A = D
Hint #5
Substitute 2×A for C, and 4×A for D in eq.4: E = 2×A + 4×A which makes E = 6×A
Hint #6
Substitute 4×A for D in eq.2: B = A + 4×A which makes B = 5×A
Hint #7
Substitute 5×A for B, and 4×A for D in eq.3: F = 5×A + 4×A which makes F = 9×A
Solution
Substitute 5×A for B, 2×A for C, 4×A for D, 6×A for E, and 9×A for F in eq.1: A + 5×A + 2×A + 4×A + 6×A + 9×A = 27 which simplifies to 27×A = 27 Divide both sides of the above equation by 27: 27×A ÷ 27 = 27 ÷ 27 which means A = 1 making B = 5×A = 5 × 1 = 5 C = 2×A = 2 × 1 = 2 D = 4×A = 4 × 1 = 4 E = 6×A = 6 × 1 = 6 F = 9×A = 9 × 1 = 9 and ABCDEF = 152469