Puzzle for March 6, 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) F = C – D eq.3) A + D = C + F eq.4) B + C – E = A + D + E + F eq.5) C = B + E eq.6)* EF = AB – D + E

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

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Hint #1


Add D to both sides of eq.2: F + D = C – D + D which becomes eq.2a) F + D = C   In eq.3, replace C with F + D (from eq.2a): A + D = F + D + F Subtract D from both sides of the above equation: A + D – D = F + D + F – D which makes A = 2×F


  

Hint #2


eq.6 may be written as: 10×E + F = 10×A + B – D + E In the equation above, add D to both sides, and subtract E from each side: 10×E + F + D – E = 10×A + B – D + E + D – E which becomes 9×E + F + D = 10×A + B Replace F + D with C (from eq.2a): eq.6a) 9×E + C = 10×A + B


  

Hint #3


In eq.6a, substitute B + E for C (from eq.5): 9×E + B + E = 10×A + B which becomes 10×E + B = 10×A + B Subtract B from both sides of the equation above: 10×E + B – B = 10×A + B – B which becomes 10×E = 10×A Divide both sides by 10: 10×E ÷ 10 = 10×A ÷ 10 which makes E = A and also makes E = A = 2×F


  

Hint #4


Add E to both sides of eq.4: B + C – E + E = A + D + E + F + E which becomes B + C = A + D + 2×E + F Substitute (2×F) for both A and E in the above equation: B + C = (2×F) + D + 2×(2×F) + F which is equivalent to B + C = 2×F + D + 4×F + F which may be written as eq.4a) B + C = F + D + 6×F


  

Hint #5


Substitute C for F + D (from eq.2a) in eq.4a: B + C = C + 6×F Subtract C from both sides of the equation above: B + C – C = C + 6×F – C which makes B = 6×F


  

Hint #6


Substitute 6×F for B, and 2×F for E in eq.5: C = 6×F + 2×F which means C = 8×F


  

Hint #7


Substitute 8×F for C in eq.2a: F + D = 8×F Subtract F from both sides: F + D – F = 8×F – F which makes D = 7×F


  

Solution

Substitute 2×F for A and E, 6×F for B, 8×F for C, and 7×F for D in eq.1: 2×F + 6×F + 8×F + 7×F + 2×F + F = 26 which simplifies to 26×F = 26 Divide both sides of the equation above by 26: 26×F ÷ 26 = 26 ÷ 26 which means F = 1 making A = E = 2×F = 2 × 1 = 2 B = 6×F = 6 × 1 = 6 C = 8×F = 8 × 1 = 8 D = 7×F = 7 × 1 = 7 and ABCDEF = 268721