Puzzle for January 17, 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 C and D to both sides of eq.3: D - C + C + D = B - D + C + D which becomes eq.3a) 2×D = B + C Add D to both sides of eq.4: A - D + D = B + D + D which becomes eq.4a) A = B + 2×D
Hint #2
In eq.4a, replace 2×D with B + C (from eq.3a): A = B + B + C which becomes eq.4b) A = 2×B + C
Hint #3
In eq.2, substitute 2×B + C for A (from eq.4b): B - C = 2×B + C - B which becomes B - C = B + C In the equation above, subtract B from both sides, and add C to both sides: B - C - B + C = B + C - B + C which makes 0 = 2×C which means 0 = C
Hint #4
Substitute 0 for C in eq.3a: 2×D = B + 0 which makes 2×D = B
Hint #5
Substitute (2×D) for B, and 0 for C in eq.4b: A = 2×(2×D) + 0 which makes A = 4×D
Hint #6
Substitute 4×D for A, and 0 for C in eq.5: E - 4×D = 4×D - 0 which becomes E - 4×D = 4×D Add 4×D to both sides of the above equation: E - 4×D + 4×D = 4×D + 4×D which makes E = 8×D
Hint #7
Substitute 4×D for A in eq.6: F - 4×D = 4×D + D which becomes F - 4×D = 5×D Add 4×D to both sides of the equation above: F - 4×D + 4×D = 5×D + 4×D which makes F = 9×D
Solution
Substitute 4×D for A, 2×D for B, 0 for C, 8×D for E, and 9×D for F in eq.1: 4×D + 2×D + 0 + D + 8×D + 9×D = 24 which simplifies to 24×D = 24 Divide both sides of the above equation by 24: 24×D ÷ 24 = 24 ÷ 24 which means D = 1 making A = 4×D = 4 × 1 = 4 B = 2×D = 2 × 1 = 2 E = 8×D = 8 × 1 = 8 F = 9×D = 9 × 1 = 9 and ABCDEF = 420189