Puzzle for September 5, 2020  ( )

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

eq.1) A + B + C + D + E + F = 15 eq.2) D + E = F eq.3) E – B + C = A – C eq.4)* C ^ F = A eq.5) A + B – C = (D × F) + C

A, B, C, D, E, and F each represent a one-digit non-negative integer.
*  "C ^ F" means "C raised to the power of F".

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Hint #1


In eq.3, add B to each side, and subtract C from each side: E – B + C + B – C = A – C + B – C which becomes eq.3a) E = A + B – 2×C   Subtract C from both sides of eq.5: A + B – C – C = (D × F) + C – C which becomes eq.5a) A + B – 2×C = (D × F)


  

Hint #2


In eq.5a, replace A + B – 2×C with E (from eq.3a): eq.5b) E = (D × F)


  

Hint #3


In eq.2, substitute (D × F) for E (from eq.5b): D + (D × F) = F which may be re-written as D × (1 + F) = F Divide both sides by (1 + F): D × (1 + F) ÷ (1 + F) = F ÷ (1 + F) which becomes eq.2a) D = F ÷ (1 + F)


  

Hint #4


To make eq.2a true, check several possible values for F and D:   If F = 0, then D = 0 ÷ (1 + 0) = 0 ÷ 1 = 0 If F = 1, then D = 1 ÷ (1 + 1) = 1 ÷ 2 = ½ If F = 2, then D = 2 ÷ (1 + 2) = 2 ÷ 3 = ⅔ If F = 3, then D = 3 ÷ (1 + 3) = 3 ÷ 4 = ¾ If F > 3, then D is a fraction (0 < D < 1)   Since D must be an integer, then D = 0 which means F = 0


  

Hint #5


Substitute 0 for D and F in eq.2: 0 + E = 0 which makes E = 0


  

Hint #6


Substitute 0 for F in eq.4: C ^ 0 = A which makes 1 = A (assumes C ≠ 0))


  

Hint #7


Substitute 0 for E, and 1 for A in eq.3: 0 – B + C = 1 – C Subtract C from each side of the above equation: 0 – B + C – C = 1 – C – C which becomes –B = 1 – 2×C Multiply each side of the equation above by (–1): –B × (–1) = (1 – 2×C) × (–1) which becomes B = –1 + 2×C which may be written as eq.3b) B = 2×C – 1


  

Solution

Substitute 1 for A, 2×C – 1 for B, and 0 for D and E and F in eq.1: 1 + 2×C – 1 + C + 0 + 0 + 0 = 15 which becomes 3×C = 15 Divide both sides of the above equation by 3: 3×C ÷ 3 = 15 ÷ 3 which means C = 5 making B = 2×C – 1 = 2×5 – 1 = 10 – 1 = 9 (from eq.3a) and ABCDEF = 195000