Puzzle for January 31, 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.5, replace A with D + E (from eq.2): B + D = D + E + E which becomes B + D = D + 2×E Subtract D from each side of the above equation: B + D – D = D + 2×E – D which makes B = 2×E
Hint #2
In eq.4, replace B with 2×E: D = 2×E + E which makes D = 3×E
Hint #3
In eq.2, substitute 3×E for D: A = 3×E + E which makes A = 4×E
Hint #4
Substitute 2×E for B, 4×E for A, and 3×E for D in eq.6: 2×E + F = 4×E + 3×E – F which becomes 2×E + F = 7×E – F In the equation above, subtract 2×E from both sides, and add F to both sides: 2×E + F – 2×E + F = 7×E – F – 2×E + F which becomes 2×F = 5×E Divide both sides by 2: 2×F ÷ 2 = 5×E ÷ 2 which makes F = 2½×E
Hint #5
Substitute 2×E for B, and 2½×E for F in eq.3: C = 2×E + 2½×E which makes C = 4½×E
Solution
Substitute 4×E for A, 2×E for B, 4½×E for C, 3×E for D, and 2½×E for F in eq.1: 4×E + 2×E + 4½×E + 3×E + E + 2½×E = 34 which simplifies to 17×E = 34 Divide both sides of the above equation by 17: 17×E ÷ 17 = 34 ÷ 17 which means E = 2 making A = 4×E = 4 × 2 = 8 B = 2×E = 2 × 2 = 4 C = 4½×E = 4½ × 2 = 9 D = 3×E = 3 × 2 = 6 F = 2½×E = 2½ × 2 = 5 and ABCDEF = 849625