Puzzle for September 28, 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
Add C to both sides of eq.2: C + D + C = B – C + C which becomes 2×C + D = B In eq.3, replace B with 2×C + D: 2×C + D = D + F Subtract D from each side of the above equation: 2×C + D – D = D + F – D which makes 2×C = F
Hint #2
In eq.4, replace F with 2×C: A = C + 2×C which makes A = 3×C
Hint #3
In eq.5, substitute 3×C for A, and 2×C for F: D = 3×C + C + 2×C which makes D = 6×C
Hint #4
Substitute 6×C for D, and 2×C for F in eq.3: B = 6×C + 2×C which makes B = 8×C
Hint #5
Substitute 6×C for D in eq.6: E = C + 6×C which makes E = 7×C
Solution
Substitute 3×C for A, 8×C for B, 6×C for D, 7×C for E, and 2×C for F in eq.1: 3×C + 8×C + C + 6×C + 7×C + 2×C = 27 which simplifies to 27×C = 27 Divide both sides of the equation above by 27: 27×C ÷ 27 = 27 ÷ 27 which means C = 1 making A = 3×C = 3 × 1 = 3 B = 8×C = 8 × 1 = 8 D = 6×C = 6 × 1 = 6 E = 7×C = 7 × 1 = 7 F = 2×C = 2 × 1 = 2 and ABCDEF = 381672