Puzzle for November 16, 2020 ( )
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 C + E (from eq.3): C + E + C = B – C + E which becomes 2×C + E = B – C + E In the above equation, subtract E from both sides, and add C to both sides: 2×C + E – E + C = B – C + E – E + C which makes 3×C = B
Hint #2
In eq.2, replace B with 3×C: 3×C + C = D which makes 4×C = D
Hint #3
In eq.6, substitute 4×C for D: 4×C – F = C + F In the equation above, add F to both sides, and subtract C from both sides: 4×C – F + F – C = C + F + F – C which becomes 3×C = 2×F Divide both sides by 2: 3×C ÷ 2 = 2×F ÷ 2 which makes 1½×C = F
Hint #4
Substitute 1½×C for F in eq.4: E = C + 1½×C which makes E = 2½×C
Hint #5
Substitute 2½×C for E in eq.3: C + 2½×C = A which makes 3½×C = A
Solution
Substitute 3½×C for A, 3×C for B, 4×C for D, 2½×C for E, and 1½×C for F in eq.1: 3½×C + 3×C + C + 4×C + 2½×C + 1½×C = 31 which simplifies to 15½×C = 31 Divide both sides of the above equation by 15½: 15½×C ÷ 15½ = 31 ÷ 15½ which means C = 2 making A = 3½×C = 3½ × 2 = 7 B = 3×C = 3 × 2 = 6 D = 4×C = 4 × 2 = 8 E = 2½×C = 2½ × 2 = 5 F = 1½×C = 1½ × 2 = 3 and ABCDEF = 762853