Puzzle for October 24, 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.5, replace D with A + B (from eq.2): A + B - C = A + C In the above equation, subtract A from both sides, and add C to both sides: A + B - C - A + C = A + C - A + C which makes B = 2×C
Hint #2
In eq.4, replace B with 2×C: 2×C = A + C Subtract C from each side of the equation above: 2×C - C = A + C - C which makes C = A
Hint #3
In eq.2, substitute C for A, and 2×C for B: D = C + 2×C which becomes D = 3×C
Hint #4
Substitute 2×C for B, C for A, and 3×C for D in eq.6: E - 2×C = C + 2×C + 3×C which becomes E - 2×C = 6×C Add 2×C to both sides of the equation above: E - 2×C + 2×C = 6×C + 2×C which makes E = 8×C
Hint #5
Substitute 8×C for E in eq.3: F = C + 8×C which makes F = 9×C
Solution
Substitute C for A, 2×C for B, 3×C for D, 8×C for E, and 9×C for F in eq.1: C + 2×C + C + 3×C + 8×C + 9×C = 24 which simplifies to 24×C = 24 Divide both sides of the above equation by 24: 24×C ÷ 24 = 24 ÷ 24 which means C = 1 making A = C = 1 B = 2×C = 2 × 1 = 2 D = 3×C = 3 × 1 = 3 E = 8×C = 8 × 1 = 8 F = 9×C = 9 × 1 = 9 and ABCDEF = 121389