Puzzle for January 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.2, replace A with C + E (from eq.4): C – E = C + E – D Subtract C from both sides of the equation above: C – E – C = C + E – D – C which becomes –E = E – D Add D and E to both sides: –E + D + E = E – D + D + E which makes D = 2×E
Hint #2
In eq.5, replace A with C + E (from eq.4): F – C = B – (C + E) + E which is equivalent to F – C = B – C – E + E which becomes F – C = B – C Add C to both sides of the above equation: F – C + C = B – C + C which makes eq.5a) F = B
Hint #3
In eq.3, substitute 2×E for D: B = C + 2×E + E which means B = C + 3×E and also means eq.3a) F = B = C + 3×E
Hint #4
Substitute 2×E for D, and C + 3×E for F (from eq.3a) in eq.6: 2×E + E + C + 3×E = A + C which becomes C + 6×E = A + C Subtract C from both sides of the above equation: C + 6×E – C = A + C – C which makes 6×E = A
Hint #5
Substitute 6×E for A in eq.4: 6×E = C + E Subtract E from both sides of the equation above: 6×E – E = C + E – E which makes 5×E = C
Hint #6
Substitute 5×E for C in eq.3a: F = B = 5×E + 3×E which makes F = B = 8×E
Solution
Substitute 6×E for A, 8×E for B and F, 5×E for C, and 2×E for D in eq.1: 6×E + 8×E + 5×E + 2×E + E + 8×E = 30 which simplifies to 30×E = 30 Divide both sides of the equation above by 30: 30×E ÷ 30 = 30 ÷ 30 which means E = 1 making A = 6×E = 6 × 1 = 6 B = F = 8×E = 8 × 1 = 8 C = 5×E = 5 × 1 = 5 D = 2×E = 2 × 1 = 2 and ABCDEF = 685218