Puzzle for September 14, 2020 ( )
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 C + D with B (from eq.4): B + E = B + F Subtract B from both sides of the equation above: B + E – B = B + F – B which makes E = F
Hint #2
In eq.3, replace F with E: D + E = C + E Subtract E from both sides of the above equation: D + E – E = C + E – E which makes D = C
Hint #3
In eq.4, substitute C for D: B = C + C which makes B = 2×C
Hint #4
Substitute 2×C for B, and C for D in eq.6: A – 2×C – C – C = 2×C + C + C which becomes A – 4×C = 4×C Add 4×C to both sides of the equation above: A – 4×C + 4×C = 4×C + 4×C which makes A = 8×C
Hint #5
Substitute 8×C for A, and 2×C for B in eq.2: C + F = 8×C + 2×C which becomes C + F = 10×C Subtract C from both sides of the equation above: C + F – C = 10×C – C which makes F = 9×C and also makes E = F = 9×C
Solution
Substitute 8×C for A, 2×C for B, C for D, and 9×C for E and F in eq.1: 8×C + 2×C + C + C + 9×C + 9×C = 30 which simplifies to 30×C = 30 Divide both sides of the above equation by 30: 30×C ÷ 30 = 30 ÷ 30 which means C = 1 making A = 8×C = 8 × 1 = 8 B = 2×C = 2 × 1 = 2 D = C = 1 E = F = 9×C = 9 × 1 = 9 and ABCDEF = 821199