Puzzle for April 3, 2020  ( )

Scratchpad

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

eq.1) A + C = B eq.2) B + C = A - C + F eq.3) A + E + F = B + C + D eq.4) F - A = A + C - E eq.5) D + E = B × C eq.6) C + D = A × C

A, B, C, D, E, and F each represent a one-digit positive integer.

Scratchpad

 

Help Area

Hint #1


In eq.2, replace B with A + C (from eq.1): A + C + C = A - C + F which becomes A + 2×C = A - C + F In the equation above, subtract A from both sides, and add C to both sides: A + 2×C - A + C = A - C + F - A + C which makes 3×C = F


  

Hint #2


In eq.4, replace F with 3×C: 3×C - A = A + C - E Subtract A and C from both sides of the above equation: 3×C - A - A - C = A + C - E - A - C which becomes 2×C - 2×A = -E Multiply both sides by (-1): 2×C - 2×A × (-1) = -E × (-1) which becomes -2×C + 2×A = E which may be written as eq.4a) 2×A - 2×C = E


  

Hint #3


In eq.3, substitute 2×A - 2×C for E (from eq.4a), 3×C for F, and A + C for B (from eq.1): A + 2×A - 2×C + 3×C = A + C + C + D which becomes 3×A + C = A + 2×C + D Subtract A and 2×C from each side of the above equation: 3×A + C - A - 2×C = A + 2×C + D - A - 2×C which becomes eq.3a) 2×A - C = D


  

Hint #4


Substitute 2×A - C for D (from eq.3a) in eq.6: C + 2×A - C = A × C which becomes 2×A = A × C Divide both sides of the above equation by A: 2×A ÷ A = A × C ÷ A which makes 2 = C making F = 3×C = 3×2 = 6


  

Hint #5


Substitute 2×A - C for D (from eq.3a), 2×A - 2×C for E (from eq.4a), and (A + C) for B in eq.5: 2×A - C + 2×A - 2×C = (A + C) × C which becomes 4×A - 3×C = C×A + C×C which is the same as eq.5a) 4×A - 3×C = A×C + C×C


  

Hint #6


Substitute 2 for C in eq.5a: 4×A - 3×2 = 2×A + 2×2 which becomes 4×A - 6 = 2×A + 4 In the above equation, subtract 2×A from both sides, and add 6 to both sides: 4×A - 6 - 2×A + 6 = 2×A + 4 - 2×A + 6 which makes 2×A = 10 Divide both sides by 2: 2×A ÷ 2 = 10 ÷ 2 which makes A = 5


  

Hint #7


Substitute 5 for A, and 2 for C in eq.1: 5 + 2 = B which makes 7 = B


  

Hint #8


Substitute 5 for A, and 2 for C in eq.3a: 2×5 - 2 = D which becomes 10 - 2 = D which makes 8 = D


  

Solution

Substitute 5 for A, and 2 for C in eq.4a: 2×5 - 2×2 = E which becomes 10 - 4 = E which makes 6 = E and ABCDEF = 572866