Puzzle for November 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.5, replace D with B + C (from eq.2): C + B + C = A + B which becomes 2×C + B = A + B Subtract B from both sides of the above equation: 2×C + B – B = A + B – B which makes 2×C = A
Hint #2
In eq.6, replace A with 2×C: E – C = 2×C + C which becomes E – C = 3×C Add C to both sides of the above equation: E – C + C = 3×C + C which makes E = 4×C
Hint #3
In eq.4, substitute 2×C for A, and 4×C for E: B – C = 2×C + 4×C which becomes B – C = 6×C Add C to both sides of the equation above: B – C + C = 6×C + C which makes B = 7×C
Hint #4
Substitute 7×C for B in eq.2: D = 7×C + C which becomes D = 8×C
Hint #5
Substitute 8×C for D in eq.3: F = C + 8×C which makes F = 9×C
Solution
Substitute 2×C for A, 7×C for B, 8×C for D, 4×C for E, and 9×C for F in eq.1: 2×C + 7×C + C + 8×C + 4×C + 9×C = 31 which simplifies to 31×C = 31 Divide both sides of the above equation by 31: 31×C ÷ 31 = 31 ÷ 31 which means C = 1 making A = 2×C = 2 × 1 = 2 B = 7×C = 7 × 1 = 7 D = 8×C = 8 × 1 = 8 E = 4×C = 4 × 1 = 4 F = 9×C = 9 × 1 = 9 and ABCDEF = 271849