Puzzle for August 1, 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.3, replace D with A + F (from eq.2): A + C = A + F + E Subtract A from each side of the equation above: A + C - A = A + F + E - A which becomes eq.3a) C = F + E
Hint #2
In eq.5, replace C with F + E (from eq.3a): F + E + E = A + F which becomes F + 2×E = A + F Subtract F from both sides of the equation above: F + 2×E - F = A + F - F which makes 2×E = A
Hint #3
In eq.4, substitute 2×E for A: F - 2×E = 2×E + E which becomes F - 2×E = 3×E Add 2×E to both sides of the above equation: F - 2×E + 2×E = 3×E + 2×E which makes F = 5×E
Hint #4
Substitute 5×E for F in eq.3a: C = 5×E + E which makes C = 6×E
Hint #5
Substitute 2×E for A, and 5×E for F in eq.2: D = 2×E + 5×E which makes D = 7×E
Hint #6
Substitute 5×E for F, and 7×E for D in eq.6: 5×E - 7×E = 7×E - B which becomes -2×E = 7×E - B Add 2×E and B to both sides of the equation above: -2×E + 2×E + B = 7×E - B + 2×E + B which makes B = 9×E
Solution
Substitute 2×E for A, 9×E for B, 6×E for C, 7×E for D, and 5×E for F in eq.1: 2×E + 9×E + 6×E + 7×E + E + 5×E = 30 which simplifies to 30×E = 30 Divide both sides of the above equation by 30: 30×E ÷ 30 = 30 ÷ 30 which means E = 1 making A = 2×E = 2 × 1 = 2 B = 9×E = 9 × 1 = 9 C = 6×E = 6 × 1 = 6 D = 7×E = 7 × 1 = 7 F = 5×E = 5 × 1 = 5 and ABCDEF = 296715