Puzzle for March 27, 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 B with D + F (from eq.2): D - F = D + F - D which becomes D - F = F Add F to both sides of the above equation: D - F + F = F + F which makes eq.6a) D = 2×F
Hint #2
In eq.5, replace D with 2×F: E + F = 2×F - E In the above equation, subtract F from both sides, and add E to both sides: E + F - F + E = 2×F - E - F + E which makes 2×E = F
Hint #3
In eq.6a, substitute (2×E) for F: D = 2×(2×E) which makes D = 4×E
Hint #4
Substitute 4×E for D, and 2×E for F in eq.2: B = 4×E + 2×E which makes B = 6×E
Hint #5
Substitute 4×E for D, and 2×E for F in eq.4: A = 4×E + E + 2×E which makes A = 7×E
Hint #6
Substitute 7×E for A, and 2×E for F in eq.3: C = 7×E + 2×E which makes C = 9×E
Solution
Substitute 7×E for A, 6×E for B, 9×E for C, 4×E for D, and 2×E for F in eq.1: 7×E + 6×E + 9×E + 4×E + E + 2×E = 29 which simplifies to 29×E = 29 Divide both sides of the above equation by 29: 29×E ÷ 29 = 29 ÷ 29 which means E = 1 making A = 7×E = 7 × 1 = 7 B = 6×E = 6 × 1 = 6 C = 9×E = 9 × 1 = 9 D = 4×E = 4 × 1 = 4 F = 2×E = 2 × 1 = 2 and ABCDEF = 769412