Puzzle for August 26, 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 C + F with B + E (from eq.2): eq.4a) A + B + E = D
Hint #2
In eq.4a, replace A + B with D + E (from eq.3): D + E + E = D which becomes D + 2×E = D Subtract D from each side of the above equation: D + 2×E – D = D – D which makes 2×E = 0 which means E = 0
Hint #3
Add C to both sides of eq.5: D – C + C = B + C + C which becomes D = B + 2×C In eq.3, substitute B + 2×C for D, and 0 for E: B + 2×C + 0 = A + B Subtract B from both sides of the equation above: B + 2×C + 0 – B = A + B – B which makes 2×C = A
Hint #4
Substitute 2×C for A in eq.4: 2×C + C + F = D which becomes eq.4b) 3×C + F = D
Hint #5
Substitute 2×C for A, and 3×C + F for D (from eq.4b) in eq.6: B + F – 2×C = 2×C + 3×C + F which becomes B + F – 2×C = 5×C + F In the equation above, subtract F from both sides, and add 2×C to both sides: B + F – 2×C – F + 2×C = 5×C + F – F + 2×C which makes B = 7×C
Hint #6
Substitute 2×C for A, 7×C for B, and 0 for E in eq.4a: 2×C + 7×C + 0 = D which makes 9×C = D
Hint #7
Substitute 7×C for B, and 0 for E in eq.2: 7×C + 0 = C + F Subtract C from each side of the equation above: 7×C + 0 – C = C + F – C which makes 6×C = F
Solution
Substitute 2×C for A, 7×C for B, 9×C for D, 0 for E, and 6×C for F in eq.1: 2×C + 7×C + C + 9×C + 0 + 6×C = 25 which simplifies to 25×C = 25 Divide both sides of the equation above by 25: 25×C ÷ 25 = 25 ÷ 25 which means C = 1 making A = 2×C = 2 × 1 = 2 B = 7×C = 7 × 1 = 7 D = 9×C = 9 × 1 = 9 F = 6×C = 6 × 1 = 6 and ABCDEF = 271906