Puzzle for October 3, 2022 ( )
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.6, replace F with C + D (from eq.4): A + C = D + C + D which becomes A + C = 2×D + C Subtract C from each side of the above equation: A + C - C = 2×D + C - C which becomes A = 2×D
Hint #2
In eq.3, replace A with 2×D: E = 2×D + D which makes E = 3×D
Hint #3
In eq.5, substitute 2×D for A, and 3×D for E: B = 2×D + 3×D which makes B = 5×D
Hint #4
Substitute 5×D for B, and 3×D for E in eq.2: C = 5×D + 3×D which makes C = 8×D
Hint #5
Substitute 8×D for C in eq.4: F = 8×D + D which makes F = 9×D
Solution
Substitute 2×D for A, 5×D for B, 8×D for C, 3×D for E, and 9×D for F in eq.1: 2×D + 5×D + 8×D + D + 3×D + 9×D = 28 which simplifies to 28×D = 28 Divide both sides of the above equation by 28: 28×D ÷ 28 = 28 ÷ 28 which means D = 1 making A = 2×D = 2 × 1 = 2 B = 5×D = 5 × 1 = 5 C = 8×D = 8 × 1 = 8 E = 3×D = 3 × 1 = 3 F = 9×D = 9 × 1 = 9 and ABCDEF = 258139