Puzzle for May 3, 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
Add E to both sides of eq.4: B - E + E = D + E + E which becomes eq.4a) B = D + 2×E Add E to both sides of eq.6: D - E + E = C + E + E which becomes eq.6a) D = C + 2×E
Hint #2
In eq.4a, replace D with C + 2×E (from eq.6a): B = C + 2×E + 2×E which becomes eq.4b) B = C + 4×E
Hint #3
In eq.3, substitute C + 4×E for B (from eq.4b): A = C + 4×E + E which becomes eq.3a) A = C + 5×E
Hint #4
Substitute C + 2×E for D (from eq.6a), and C + 5×E for A (from eq.3a) in eq.5: C + C + 2×E = C + 5×E + E which becomes 2×C + 2×E = C + 6×E Subtract 2×E and C from each side of the above equation: 2×C + 2×E - 2×E - C = C + 6×E - 2×E - C which makes C = 4×E
Hint #5
Substitute 4×E for C in eq.3a: A = 4×E + 5×E which makes A = 9×E
Hint #6
Substitute 4×E for C in eq.4b: B = 4×E + 4×E which makes B = 8×E
Hint #7
Substitute 4×E for C in eq.6a: D = 4×E + 2×E which makes D = 6×E
Hint #8
Substitute 4×E for C in eq.2: 4×E = E + F Subtract E from each side of the above equation: 4×E - E = E + F - E which makes 3×E = F
Solution
Substitute 9×E for A, 8×E for B, 4×E for C, 6×E for D, and 3×E for F in eq.1: 9×E + 8×E + 4×E + 6×E + E + 3×E = 31 which simplifies to 31×E = 31 Divide both sides of the above equation by 31: 31×E ÷ 31 = 31 ÷ 31 which means E = 1 making A = 9×E = 9 × 1 = 9 B = 8×E = 8 × 1 = 8 C = 4×E = 4 × 1 = 4 D = 6×E = 6 × 1 = 6 F = 3×E = 3 × 1 = 3 and ABCDEF = 984613