Puzzle for April 5, 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 F with C + D (from eq.2): D + E = C + D - E In the above equation, subtract D from both sides, and add E to both sides: D + E - D + E = C + D - E - D + E which becomes eq.3a) 2×E = C
Hint #2
In eq.6, replace F with C + D (from eq.2): C + E + C + D = A + B + D which becomes 2×C + E + D = A + B + D Subtract D from each side of the above equation: 2×C + E + D - D = A + B + D - D which becomes eq.6a) 2×C + E = A + B
Hint #3
In eq.6a, substitute (2×E) for C: 2×(2×E) + E = A + B which becomes 4×E + E = A + B which becomes 5×E = A + B Subtract B from each side of the equation above: 5×E - B = A + B - B which becomes eq.6b) 5×E - B = A
Hint #4
Substitute 5×E - B for A (from eq.6b) in eq.5: 5×E - B + E = B - E which becomes 6×E - B = B - E Add B and E to both sides of the above equation: 6×E - B + B + E = B - E + B + E which becomes 7×E = 2×B Divide both sides by 2: 7×E ÷ 2 = 2×B ÷ 2 which makes 3½×E = B
Hint #5
Substitute 3½×E for B in eq.6b: 5×E - 3½×E = A which makes 1½×E = A
Hint #6
Substitute 3½×E for B, and 1½×E for A in eq.4: 3½×E - D = D - 1½×E Add D and 1½×E to both sides of the above equation: 3½×E - D + D + 1½×E = D - 1½×E + D + 1½×E which makes 5×E = 2×D Divide both sides by 2: 5×E ÷ 2 = 2×D ÷ 2 which makes 2½×E = D
Hint #7
Substitute 2×E for C, and 2½×E for D in eq.2: F = 2×E + 2½×E which makes F = 4½×E
Solution
Substitute 1½×E for A, 3½×E for B, 2×E for C, 2½×E for D, and 4½×E for F in eq.1: 1½×E + 3½×E + 2×E + 2½×E + E + 4½×E = 30 which simplifies to 15×E = 30 Divide both sides of the above equation by 15: 15×E ÷ 15 = 30 ÷ 15 which means E = 2 making A = 1½×E = 1½ × 2 = 3 B = 3½×E = 3½ × 2 = 7 C = 2×E = 2 × 2 = 4 D = 2½×E = 2½ × 2 = 5 F = 4½×E = 4½ × 2 = 9 and ABCDEF = 374529