Puzzle for April 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
In eq.6, replace F with C + E (from eq.3): B + E = C + C + E which becomes B + E = 2×C + E Subtract E from both sides of the above equation: B + E – E = 2×C + E – E which makes eq.6a) B = 2×C
Hint #2
In eq.2, replace B with 2×C: 2×C + C = D which makes eq.2a) 3×C = D
Hint #3
In eq.4, substitute 3×C for D, B + E for A (from eq.5), and C + E for F (from eq.3): B + 3×C = B + E + E + C + E which becomes B + 3×C = B + C + 3×E Subtract B and C from both sides of the equation above: B + 3×C – B – C = B + C + 3×E – B – C which makes 2×C = 3×E Divide both sides by 2: 2×C ÷ 2 = 3×E ÷ 2 which makes C = 1½×E
Hint #4
Substitute (1½×E) for C in eq.6a: B = 2×(1½×E) which makes B = 3×E
Hint #5
Substitute (1½×E) for C in eq.2a: 3×(1½×E) = D which makes 4½×E = D
Hint #6
Substitute 3×E for B in eq.5: A = 3×E + E which makes A = 4×E
Hint #7
Substitute 1½×E for C in eq.3: F = 1½×E + E which makes F = 2½×E
Solution
Substitute 4×E for A, 3×E for B, 1½×E for C, 4½×E for D, and 2½×E for F in eq.1: 4×E + 3×E + 1½×E + 4½×E + E + 2½×E = 33 which simplifies to 16½×E = 33 Divide both sides of the equation above by 16½: 16½×E ÷ 16½ = 33 ÷ 16½ which means E = 2 making A = 4×E = 4 × 2 = 8 B = 3×E = 3 × 2 = 6 C = 1½×E = 1½ × 2 = 3 D = 4½×E = 4½ × 2 = 9 F = 2½×E = 2½ × 2 = 5 and ABCDEF = 863925