Puzzle for September 28, 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
Subtract A from both sides of eq.3: C - A - A = A + E - A which becomes eq.3a) C - 2×A = E
Hint #2
In eq.5, replace E with C - 2×A (from eq.3a): C - 2×A - C = C - B which becomes -2×A = C - B Add 2×A and B to both sides of the above equation: -2×A + 2×A + B = C - B + 2×A + B which becomes eq.5a) B = C + 2×A
Hint #3
In eq.4, replace B with C + 2×A (from eq.5a): D - C = C + 2×A - D Add C and D to both sides of the above equation: D - C + C + D = C + 2×A - D + C + D which becomes 2×D = 2×C + 2×A Divide both sides by 2: 2×D ÷ 2 = (2×C + 2×A) ÷ 2 which becomes eq.4a) D = C + A
Hint #4
In eq.2, substitute (C + 2×A) for B (from eq.5a), and (C + A) for D (from eq.4a): (C + 2×A) - (C + A) = F - (C + 2×A) which becomes C + 2×A - C - A = F - C - 2×A which becomes A = F - C - 2×A Add C and 2×A to both sides of the above equation: A + C + 2×A = F - C - 2×A + C + 2×A which becomes eq.2a) C + 3×A = F
Hint #5
Substitute C + 3×A for F (from eq.2a) into eq.6: C + 3×A - C = C - A which becomes 3×A = C - A Add A to both sides of the equation above: 3×A + A = C - A + A which makes 4×A = C
Hint #6
Substitute 4×A for C in eq.5a: B = 4×A + 2×A which makes B = 6×A
Hint #7
Substitute 4×A for C in eq.4a: D = 4×A + A which makes D = 5×A
Hint #8
Substitute 4×A for C in eq.3a: 4×A - 2×A = E which makes 2×A = E
Hint #9
Substitute 4×A for C in eq.2a: 4×A + 3×A = F which makes 7×A = F
Solution
Substitute 6×A for B, 4×A for C, 5×A for D, 2×A for E, and 7×A for F in eq.1: A + 6×A + 4×A + 5×A + 2×A + 7×A = 25 which simplifies to 25×A = 25 Divide both sides of the above equation by 25: 25×A ÷ 25 = 25 ÷ 25 which makes A = 1 making B = 6×A = 6 × 1 = 6 C = 4×A = 4 × 1 = 4 D = 5×A = 5 × 1 = 5 E = 2×A = 2 × 1 = 2 F = 7×A = 7 × 1 = 7 and ABCDEF = 164527