Puzzle for September 3, 2020  ( )

Scratchpad

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

eq.1) A = B + E eq.2) C – D = B – E eq.3) D – E – F = A – C eq.4) C + D = A + B + F eq.5) B × E = A × F eq.6) log [base F] (E) = A ÷ B

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

Scratchpad

 

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Hint #1


Add E, F, and C to both sides of eq.3: D – E – F + E + F + C = A – C + E + F + C which becomes D + C = A + E + F which is the same as eq.3a) C + D = A + E + F   In eq.4, replace C + D with A + E + F (from eq.3a): A + E + F = A + B + F Subtract A and F from both sides of the equation above: A + E + F – A – F = A + B + F – A – F which simplifies to E = B


  

Hint #2


In eq.2, replace E with B: C – D = B – B which makes C – D = 0 Add D to both sides of the equation above: C – D + D = 0 + D which makes C = D


  

Hint #3


In eq.1, replace E with B: A = B + B which makes eq.1a) A = 2×B


  

Hint #4


In eq.5, substitute B for E, and 2×B for A: B × B = 2×B × F Divide both sides of the equation above by B: (B × B) ÷ B = (2×B × F) ÷ B which means B = 2×F and also means E = B = 2×F


  

Hint #5


Substitute (2×F) for B in eq.1a: A = 2×(2×F) which makes A = 4×F


  

Hint #6


Substitute C for D, 4×F for A, and 2×F for E in eq.3a: C + C = 4×F + 2×F + F which makes 2×C = 7×F Divide both sides of the above equation by 2: 2×C ÷ 2 = 7×F ÷ 2 which makes C = 3½×F and also makes D = C = 3½×F


  

Hint #7


eq.6 is a logarithmic equation, and may be re-written as an exponential equation: eq.6a) F ^ (A ÷ B) = E (In eq.6a above, "F ^ (A ÷ B)" means "F raised to the power of (A ÷ B)".)


  

Solution

In eq.6a, substitute 2×B for A, and 2×F for E: F ^ (2×B ÷ B) = 2×F which becomes F ^ 2 = 2×F which may be written as F × F = 2×F Divide both sides of the equation above by F: (F × F) ÷ F = 2×F ÷ F which makes F = 2 making A = 4×F = 4 × 2 = 8 B = E = 2×F = 2 × 2 = 4 C = D = 3½×F = 3½ × 2 = 7 and ABCDEF = 847742