Puzzle for January 3, 2020  ( )

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Find the 6-digit number ABCDEF by solving the following equations:

eq.1) E = C – F eq.2) A + F = B eq.3) B + F = A × D eq.4) C + F = B + E eq.5)* AB = (C + D – B) × (A × F) eq.6)** D! = A + B

A, B, C, D, E, and F each represent a one-digit positive integer.
*  AB is a 2-digit number (not A×B).
**  D! is D-factorial.

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Hint #1


In eq.4, replace E with C – F (from eq.1): C + F = B + C – F In the above equation, add F to both sides, and subtract C from each side: C + F + F – C = B + C – F + F – C which simplifies to eq.4a) B = 2×F


  

Hint #2


In eq.2, replace B with 2×F: A + F = 2×F Subtract F from each side of the equation above: A + F – F = 2×F – F which makes A = F


  

Hint #3


In eq.3, substitute 2×F for B, and F for A: 2×F + F = F × D which becomes 3×F = F × D Divide both sides of the equation above by F: 3×F ÷ F = F × D ÷ F which makes 3 = D


  

Hint #4


Substitute 3 for D, F for A, and 2×F for B in eq.6: 3! = F + 2×F which becomes 3 × 2 × 1 = 3×F which makes 6 = 3×F Divide both sides of the above equation by 3: 6 ÷ 3 = 3×F ÷ 3 which means 2 = F which also means A = F = 2


  

Hint #5


Substitute 2 for F in eq.4a: B = 2×2 which makes B = 4


  

Solution

eq.5 may be written as: 10×A + B = (C + D – B) × (A × F) Substitute 2 for A and F, 4 for B, and 3 for D in the equation above: 10×2 + 4 = (C + 3 – 4) × (2 × 2) which becomes 24 = (C – 1) × 4 Divide both sides by 4: 24 ÷ 4 = (C – 1) × 4 ÷ 4 which becomes 6 = C – 1 Add 1 to both sides: 6 + 1 = C – 1 + 1 which means 7 = C making E = C – F = 7 – 2 = 5 (from eq.1) and ABCDEF = 247352