Puzzle for November 12, 2019 ( )
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 A and D to both sides of eq.5: B – A + A + D = E – D + A + D which becomes B + D = E + A In eq.6, replace B + D with E + A: A + E + A = E + F Subtract E from both sides of the above equations: A + E + A – E = E + F – E which means 2×A = F
Hint #2
In eq.3, replace F with 2×A: E – A = D + 2×A In the above equation, add A to each side, and subtract D from both sides: E – A + A – D = D + 2×A + A – D which becomes eq.3a) E – D = 3×A
Hint #3
In eq.5, substitute 3×A for E – D (from eq.3a): B – A = 3×A Add A to both sides of the above equations: B – A + A = 3×A + A which makes B = 4×A
Hint #4
Substitute 2×A for F, and 4×A for B in eq.2: 2×A = D – 4×A Add 4×A to both sides of the above equation: 2×A + 4×A = D – 4×A + 4×A which makes 6×A = D
Hint #5
Substitute 6×A for D in eq.3a: E – 6×A = 3×A Add 6×A to both sides of the equation above: E – 6×A + 6×A = 3×A + 6×A which makes E = 9×A
Hint #6
Substitute 6×A for D in eq.4: 6×A = A + C Subtract A from both sides of the above equation: 6×A – A = A + C – A which becomes 5×A = C
Solution
Substitute 4×A for B, 5×A for C, 6×A for D, 9×A for E, and 2×A for F in eq.1: A + 4×A + 5×A + 6×A + 9×A + 2×A = 27 which simplifies to 27×A = 27 Divide both sides of the equation above by 27: 27×A ÷ 27 = 27 ÷ 27 which means A = 1 making B = 4×A = 4 × 1 = 4 C = 5×A = 5 × 1 = 5 D = 6×A = 6 × 1 = 6 E = 9×A = 9 × 1 = 9 F = 2×A = 2 × 1 = 2 and ABCDEF = 145692