Puzzle for December 20, 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
Add D to both sides of eq.6: A + D + D = C – D + D which becomes eq.6a) A + 2×D = C In eq.5, replace C with A + 2×D (from eq.6a): E = A + 2×D + D – A which makes E = 3×D
Hint #2
In eq.2, replace E with 3×D: F = D + 3×D which makes F = 4×D
Hint #3
In eq.4, replace E with 3×D, and F with 4×D: A = 3×D + 4×D which makes A = 7×D
Hint #4
Substitute 7×D for A in eq.3: B = 7×D + D which makes B = 8×D
Hint #5
Substitute 7×D for A in eq.6a: 7×D + 2×D = C which makes 9×D = C
Solution
Substitute 7×D for A, 8×D for B, 9×D for C, 3×D for E, and 4×D for F in eq.1: 7×D + 8×D + 9×D + D + 3×D + 4×D = 32 which simplifies to 32×D = 32 Divide both sides of the above equation by 32: 32×D ÷ 32 = 32 ÷ 32 which means D = 1 making A = 7×D = 7 × 1 = 7 B = 8×D = 8 × 1 = 8 C = 9×D = 9 × 1 = 9 E = 3×D = 3 × 1 = 3 F = 4×D = 4 × 1 = 4 and ABCDEF = 789134