Puzzle for November 6, 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
In eq.6, replace A with C + E (from eq.4): D + E = C + E + C which becomes D + E = 2×C + E Subtract E from each side of the equation above: D + E - E = 2×C + E - E which makes D = 2×C
Hint #2
In eq.2, replace D with 2×C: E = C + 2×C which makes E = 3×C
Hint #3
In eq.4, substitute 3×C for E: A = C + 3×C which makes A = 4×C
Hint #4
Substitute 4×C for A, and 3×C for E in eq.5: B = 4×C + C + 3×C which makes B = 8×C
Hint #5
Substitute 8×C for B in eq.3: F = 8×C + C which makes F = 9×C
Solution
Substitute 4×C for A, 8×C for B, 2×C for D, 3×C for E, and 9×C for F in eq.1: 4×C + 8×C + C + 2×C + 3×C + 9×C = 27 which simplifies to 27×C = 27 Divide both sides of the above equation by 27: 27×C ÷ 27 = 27 ÷ 27 which means C = 1 making A = 4×C = 4 × 1 = 4 B = 8×C = 8 × 1 = 8 D = 2×C = 2 × 1 = 2 E = 3×C = 3 × 1 = 3 F = 9×C = 9 × 1 = 9 and ABCDEF = 481239