Puzzle for August 7, 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 D to both sides of eq.6: C - D + D = D + F + D which becomes eq.6a) C = 2×D + F
Hint #2
In eq.5, replace C with 2×D + F: F - D = 2×D + F - F which becomes F - D = 2×D Add D to both sides: F - D + D = 2×D + D which makes F = 3×D
Hint #3
In eq.6a, replace F with 3×D: C = 2×D + 3×D which makes C = 5×D
Hint #4
In eq.3, substitute 5×D for C: B = 5×D + D which makes B = 6×D
Hint #5
Substitute 5×D for C, and 3×D for F in eq.2: E = 5×D + 3×D which makes E = 8×D
Hint #6
Substitute 8×D for E eq.4: A = D + 8×D which makes A = 9×D
Solution
Substitute 9×D for A, 6×D for B, 5×D for C, 8×D for E, and 3×D for F in eq.1: 9×D + 6×D + 5×D + D + 8×D + 3×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 = 9×D = 9 × 1 = 9 B = 6×D = 6 × 1 = 6 C = 5×D = 5 × 1 = 5 E = 8×D = 8 × 1 = 8 F = 3×D = 3 × 1 = 3 and ABCDEF = 965183