Puzzle for April 14, 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.
* DE is a 2-digit number (not D×E).
Scratchpad
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Hint #1
In eq.6, replace A + B - D with D + E + F (from eq.4): DE - C = D + E + F + C which may be written as 10×D + E - C = D + E + F + C Subtract E, D, and C from both sides of the above equation: 10×D + E - C - E - D - C = D + E + F + C - E - D - C which becomes eq.6a) 9×D - 2×C = F
Hint #2
In eq.3, replace F with 9×D - 2×C (from eq.6a): C + 9×D - 2×C = B + D which becomes 9×D - C = B + D Subtract D from each side of the above equation: 9×D - C - D = B + D - D which becomes eq.3a) 8×D - C = B
Hint #3
In eq.2, substitute 8×D - C for B (from eq.3a): A = 8×D - C + C which makes A = 8×D
Hint #4
Substitute 9×D - 2×C for F (from eq.6a), 8×D for A, and 8×D - C for B (from eq.3a) in eq.4: D + E + 9×D - 2×C = 8×D + 8×D - C - D which becomes 10×D + E - 2×C = 15×D - C In the above equation, subtract 10×D from both sides, and add 2×C to both sides: 10×D + E - 2×C - 10×D + 2×C = 15×D - C - 10×D + 2×C which becomes eq.4a) E = 5×D + C
Hint #5
Substitute 8×D - C for B (from eq.3a), (5×D + C) for E (from eq.4a), 9×D - 2×C for F (from eq.6a), and 8×D for A in eq.5: 8×D - C - (5×D + C) + 9×D - 2×C = 8×D - D + (5×D + C) which becomes 8×D - C - 5×D - C + 9×D - 2×C = 8×D - D + 5×D + C which becomes 12×D - 4×C = 12×D + C In the above equation, subtract 12×D from both sides, and add 4×C to both sides: 12×D - 4×C - 12×D + 4×C = 12×D + C - 12×D + 4×C which makes 0 = 5×C which means 0 = C
Hint #6
Substitute 0 for C in eq.3a: 8×D - 0 = B which makes 8×D = B
Hint #7
Substitute 0 for C in eq.4a: E = 5×D + 0 which makes E = 5×D
Hint #8
Substitute 0 for C in eq.6a: 9×D - 2×0 = F which becomes 9×D - 0 = F which makes 9×D = F
Solution
Substitute 8×D for A and B, 0 for C, 5×D for E, and 9×D for F in eq.1: 8×D + 8×D + 0 + D + 5×D + 9×D = 31 which simplifies to 31×D = 31 Divide both sides of the above equation by 31: 31×D ÷ 31 = 31 ÷ 31 which means D = 1 making A = B = 8×D = 8 × 1 = 8 E = 5×D = 5 × 1 = 5 F = 9×D = 9 × 1 = 9 and ABCDEF = 880159