Puzzle for July 1, 2020  ( )

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

eq.1) A + B + C + D + E + F = 26 eq.2) B = A + C eq.3) A + E = B + D eq.4) D – F = B + C – D eq.5) C + D = E + F eq.6)* AB = A + B + C + D + E – F

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

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Hint #1


In eq.3, replace B with A + C (from eq.2): A + E = A + C + D Subtract A from each side of the equation above: A + E – A = A + C + D – A which becomes eq.3a) E = C + D


  

Hint #2


In eq.5, replace C + D with E (from eq.3a): E = E + F Subtract E from both sides of the equation above: E – E = E + F – E which makes 0 = F


  

Hint #3


In eq.1, substitute 0 for F: A + B + C + D + E + 0 = 26 which becomes eq.1a) A + B + C + D + E = 26


  

Hint #4


In eq.6, substitute 26 for A + B + C + D + E (from eq.1a), and substitute 0 for F: AB = 26 – 0 which means AB = 26 and therefore means A = 2 and B = 6


  

Hint #5


Substitute 6 for B, and 2 for A in eq.2: 6 = 2 + C Subtract 2 from both sides of the equation above: 6 – 2 = 2 + C – 2 which makes 4 = C


  

Hint #6


Substitute 0 for F, 6 for B, and 4 for C in eq.4: D – 0 = 6 + 4 – D which becomes D = 10 – D Add D to both sides of the equation above: D + D = 10 – D + D which makes 2×D = 10 Divide both sides by 2: 2×D ÷ 2 = 10 ÷ 2 which makes D = 5


  

Solution

Substitute 4 for C, 5 for D, and 0 for F in eq.5: 4 + 5 = E + 0 which makes 9 = E making ABCDEF = 264590