Puzzle for December 21, 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.3, replace D with A + C (from eq.2): A + B = A + C + E Subtract A from each side of the equation above: A + B - A = A + C + E - A which becomes eq.3a) B = C + E
Hint #2
In eq.5, replace B with C + E (from eq.3a): E + F = C + E + C + D which becomes E + F = 2×C + E + D Subtract E from each side of the above equation: E + F - E = 2×C + E + D - E which becomes eq.5a) F = 2×C + D
Hint #3
In eq.4, substitute 2×C + D for F (from eq.5a): D + E = C + 2×C + D which becomes D + E = 3×C + D Subtract D from both sides of the above equation: D + E - D = 3×C + D - D which makes E = 3×C
Hint #4
Substitute 3×C for E in eq.3a: B = C + 3×C which makes B = 4×C
Hint #5
eq.6 may be re-written as: D + F = A + C + B + E In the equation above, substitute D for A + C (from eq.2), 4×C for B, and 3×C for E: D + F = D + 4×C + 3×C which becomes D + F = D + 7×C Subtract D from both sides of the above equation: D + F - D = D + 7×C - D which makes F = 7×C
Hint #6
Substitute 3×C for E, and 7×C for F in eq.4: D + 3×C = C + 7×C which becomes D + 3×C = 8×C Subtract 3×C from both sides of the equation above: D + 3×C - 3×C = 8×C - 3×C which makes D = 5×C
Hint #7
Substitute 5×C for D in eq.2: 5×C = A + C Subtract C from each side of the equation above: 5×C - C = A + C - C which makes 4×C = A
Solution
Substitute 4×C for A and B, 5×C for D, 3×C for E, and 7×C for F in eq.1: 4×C + 4×C + C + 5×C + 3×C + 7×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 = B = 4×C = 4 × 1 = 4 D = 5×C = 5 × 1 = 5 E = 3×C = 3 × 1 = 3 F = 7×C = 7 × 1 = 7 and ABCDEF = 441537