Puzzle for November 3, 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.4, replace E with C + D (from eq.2): A + C + C + D = B + D which becomes A + 2×C + D = B + D Subtract D from each side of the equation above: A + 2×C + D – D = B + D – D which becomes eq.4a) A + 2×C = B
Hint #2
In eq.3, replace B with A + 2×C (from eq.4a): A + 2×C + C = A + F which becomes A + 3×C = A + F Subtract A from each side of the equation above: A + 3×C – A = A + F – A which makes 3×C = F
Hint #3
In eq.5, substitute 3×C for F: eq.5a) E + 3×C = B
Hint #4
Substitute E + 3×C for B (from eq.5a), and 3×C for F in eq.6: E + 3×C + 3×C = A + E which becomes E + 6×C = A + E Subtract E from both sides of the equation above: E + 6×C – E = A + E – E which makes 6×C = A
Hint #5
Substitute 6×C for A, and 3×C for F in eq.3: B + C = 6×C + 3×C which becomes B + C = 9×C Subtract C from each side of the equation above: B + C – C = 9×C – C which makes B = 8×C
Hint #6
Substitute 8×C for B into eq.5a: E + 3×C = 8×C Subtract 3×C from each side of the above equation: E + 3×C – 3×C = 8×C – 3×C which makes E = 5×C
Hint #7
Substitute 5×C for E in eq.2: C + D = 5×C Subtract C from both sides of the above equation: C + D – C = 5×C – C which makes D = 4×C
Solution
Substitute 6×C for A, 8×C for B, 4×C for D, 5×C for E, and 3×C for F in eq.1: 6×C + 8×C + C + 4×C + 5×C + 3×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 = 6×C = 6 × 1 = 6 B = 8×C = 8 × 1 = 8 D = 4×C = 4 × 1 = 4 E = 5×C = 5 × 1 = 5 F = 3×C = 3 × 1 = 3 and ABCDEF = 681453