Puzzle for August 31, 2021 ( )
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 F with C + E (from eq.4): B + E = C + C + E which becomes B + E = 2×C + E Subtract E from each side of the above equation: B + E – E = 2×C + E – E which makes B = 2×C
Hint #2
In eq.3, replace B with 2×C: D = 2×C + C which makes D = 3×C
Hint #3
In eq.6, substitute 2×C for B, and 3×C for D: E + F = A + 2×C + 3×C which becomes eq.6a) E + F = A + 5×C
Hint #4
Substitute C + F for A (from eq.2) in eq.6a: E + F = C + F + 5×C which becomes E + F = 6×C + F Subtract F from both sides of the above equation: E + F – F = 6×C + F – F which makes E = 6×C
Hint #5
Substitute 2×C for B, and 6×C for E in eq.5: 2×C + 6×C = C + F which becomes 8×C = C + F Subtract C from each side of the equation above: 8×C – C = C + F – C which makes 7×C = F
Hint #6
Substitute 7×C for F in eq.2: C + 7×C = A which makes 8×C = A
Solution
Substitute 8×C for A, 2×C for B, 3×C for D, 6×C for E, and 7×C for F in eq.1: 8×C + 2×C + C + 3×C + 6×C + 7×C = 27 which simplifies to 27×C = 27 Divide both sides of the above equation by 27: 27×C ÷ 27 = 27 ÷ 27 which means C = 1 making A = 8×C = 8 × 1 = 8 B = 2×C = 2 × 1 = 2 D = 3×C = 3 × 1 = 3 E = 6×C = 6 × 1 = 6 F = 7×C = 7 × 1 = 7 and ABCDEF = 821367