Puzzle for September 28, 2022 ( )
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.2, replace F with B - E (from eq.6): B = D - E + B - E which becomes B = D - 2×E + B In the above equation, subtract B from both sides, and add 2×E to both sides: B - B + 2×E = D - 2×E + B - B + 2×E which simplifies to eq.2a) 2×E = D
Hint #2
In eq.4, replace D with 2×E: 2×E = A - C Add C to both sides of the equation above: 2×E + C = A - C + C which becomes eq.4a) 2×E + C = A
Hint #3
Add E to both sides of eq.5: E + E = C - E + F + E which becomes 2×E = C + F In eq.4a, replace 2×E with C + F: C + F + C = A which becomes eq.4b) 2×C + F = A
Hint #4
In eq.3, substitute (2×C + F) for A (from eq.4b): C = E + F - (2×C + F) which becomes C = E + F - 2×C - F which becomes C = E - 2×C Add 2×C to both sides of the above equation: C + 2×C = E - 2×C + 2×C which makes 3×C = E
Hint #5
Substitute (3×C) for E in eq.2a: 2×(3×C) = D which makes 6×C = D
Hint #6
Substitute (3×C) for E in eq.4a: 2×(3×C) + C = A which becomes 6×C + C = A which makes 7×C = A
Hint #7
Substitute 7×C for A in eq.4b: 2×C + F = 7×C Subtract 2×C from each side of the above equation: 2×C + F - 2×C = 7×C - 2×C which makes F = 5×C
Hint #8
Substitute 5×C for F, and 3×C for E in eq.6: 5×C = B - 3×C Add 3×C to both sides of the above equation: 5×C + 3×C = B - 3×C + 3×C which makes 8×C = B
Solution
Substitute 7×C for A, 8×C for B, 6×C for D, 3×C for E, and 5×C for F in eq.1: 7×C + 8×C + C + 6×C + 3×C + 5×C = 30 which simplifies to 30×C = 30 Divide both sides of the above equation by 30: 30×C ÷ 30 = 30 ÷ 30 which means C = 1 making A = 7×C = 7 × 1 = 7 B = 8×C = 8 × 1 = 8 D = 6×C = 6 × 1 = 6 E = 3×C = 3 × 1 = 3 F = 5×C = 5 × 1 = 5 and ABCDEF = 781635