Puzzle for April 7, 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
Add D to both sides of eq.5: E – D + D = A + D + D which becomes E = A + 2×D In eq.4, replace E with A + 2×D: D + A + 2×D = A + B which becomes A + 3×D = A + B Subtract A from both sides of the equation above: A + 3×D – A = A + B – A which makes 3×D = B
Hint #2
In eq.6, replace B with 3×D: 3×D = D + F Subtract D from both sides of the above equation: 3×D – D = D + F – D which makes 2×D = F
Hint #3
In eq.4, replace A with C + D (from eq.3): D + E = C + D + B Subtract D from each side of the above equation: D + E – D = C + D + B – D which becomes eq.4a) E = C + B
Hint #4
In eq.2, substitute C + B for E (from eq.4a): B + C + B = A + C which becomes 2×B + C = A + C Subtract C from each side of the equation above: 2×B + C – C = A + C – C which means eq.2a) 2×B = A
Hint #5
Substitute (3×D) for B in eq.2a: 2×(3×D) = A which makes 6×D = A
Hint #6
Substitute 6×D for A in eq.3: C + D = 6×D Subtract D from both sides of the above equation: C + D – D = 6×D – D which makes C = 5×D
Hint #7
Substitute 5×D for C, and 3×D for B in eq.4a: E = 5×D + 3×D which makes E = 8×D
Solution
Substitute 6×D for A, 3×D for B, 5×D for C, 8×D for E, and 2×D for F in eq.1: 6×D + 3×D + 5×D + D + 8×D + 2×D = 25 which simplifies to 25×D = 25 Divide both sides of the equation above by 25: 25×D ÷ 25 = 25 ÷ 25 which means D = 1 making A = 6×D = 6 × 1 = 6 B = 3×D = 3 × 1 = 3 C = 5×D = 5 × 1 = 5 E = 8×D = 8 × 1 = 8 F = 2×D = 2 × 1 = 2 and ABCDEF = 635182