Puzzle for November 23, 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 A with D + E (from eq.2): D + E + E = C – E which becomes D + 2×E = C – E Add E to both sides of the above equation: D + 2×E + E = C – E + E which becomes eq.5a) D + 3×E = C
Hint #2
In eq.4, replace C with D + 3×E (from eq.5a): D + 3×E = D + E + F Subtract D and E from both sides of the above equation: D + 3×E – D – E = D + E + F – D – E which simplifies to eq.4a) 2×E = F
Hint #3
In eq.6, substitute 2×E for F: D + E = 2×E – D In the equation above, subtract E from both sides, and add D to both sides: D + E – E + D = 2×E – D – E + D which makes 2×D = E
Hint #4
Substitute (2×D) for E into eq.4a: 2×(2×D) = F which makes 4×D = F
Hint #5
Substitute (2×D) for E in eq.5a: D + 3×(2×D) = C which becomes D + 6×D = C which makes 7×D = C
Hint #6
Substitute 2×D for E in eq.2: A = D + 2×D which makes A = 3×D
Hint #7
Substitute 7×D for C in eq.3: B = 7×D + D which makes B = 8×D
Solution
Substitute 3×D for A, 8×D for B, 7×D for C, 2×D for E, and 4×D for F in eq.1: 3×D + 8×D + 7×D + D + 2×D + 4×D = 25 which simplifies to 25×D = 25 Divide both sides of the above equation by 25: 25×D ÷ 25 = 25 ÷ 25 which means D = 1 making A = 3×D = 3 × 1 = 3 B = 8×D = 8 × 1 = 8 C = 7×D = 7 × 1 = 7 E = 2×D = 2 × 1 = 2 F = 4×D = 4 × 1 = 4 and ABCDEF = 387124