Puzzle for November 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.
Scratchpad
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Hint #1
In eq.5, replace B with D + E (from eq.3): eq.5a) F = A + D + E
Hint #2
In eq.6, replace F with A + D + E (from eq.5a): E + A + D + E = A + C + D which becomes 2×E + A + D = A + C + D Subtract A and D from each side of the above equation: 2×E + A + D - A - D = A + C + D - A - D which simplifies to 2×E = C
Hint #3
In eq.4, substitute 2×E for C: D = 2×E + E which makes D = 3×E
Hint #4
Substitute 3×E for D in eq.3: B = 3×E + E which makes B = 4×E
Hint #5
Substitute 2×E for C, and 3×E for D in eq.2: A = 2×E + 3×E which makes A = 5×E
Hint #6
Substitute 5×E for A, and 4×E for B in eq.5: F = 5×E + 4×E which makes F = 9×E
Solution
Substitute 5×E for A, 4×E for B, 2×E for C, 3×E for D, and 9×E for F in eq.1: 5×E + 4×E + 2×E + 3×E + E + 9×E = 24 which simplifies to 24×E = 24 Divide both sides of the above equation by 24: 24×E ÷ 24 = 24 ÷ 24 which means E = 1 making A = 5×E = 5 × 1 = 5 B = 4×E = 4 × 1 = 4 C = 2×E = 2 × 1 = 2 D = 3×E = 3 × 1 = 3 F = 9×E = 9 × 1 = 9 and ABCDEF = 542319