Puzzle for August 11, 2022 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
eq.6 may be written as: A = (B + C + D + E + F) ÷ 5 Multiply both sides of the above equation by 5: 5 × A = 5 × (B + C + D + E + F) ÷ 5 which becomes eq.6a) 5×A = B + C + D + E + F
Hint #2
In eq.1, replace B + C + D + E + F with 5×A (from eq.6a): A + 5×A = 12 which makes 6×A = 12 Divide both sides of the above equation by 6: 6×A ÷ 6 = 12 ÷ 6 which makes A = 2
Hint #3
In eq.5, substitute (E + F) for D (from eq.2): (E + F) ÷ F = E + (E ÷ F) which may be written as (E ÷ F) + (F ÷ F) = E + (E ÷ F) which becomes (E ÷ F) + 1 = E + (E ÷ F) Subtract (E ÷ F) from each side of the equation above: (E ÷ F) + 1 – (E ÷ F) = E + (E ÷ F) – (E ÷ F) which makes 1 = E
Hint #4
Substitute 1 for E in eq.2: eq.2a) D = 1 + F
Hint #5
Substitute 1 + F for D (from eq.2a), 2 for A, and 1 for E in eq.3: 1 + F – 2 = 2 – B + 1 which becomes F – 1 = 3 – B Add 1 to both sides of the equation above: F – 1 + 1 = 3 – B + 1 which becomes eq.3a) F = 4 – B
Hint #6
Substitute 2 for A, 1 + F for D (from eq.2a), and 1 for E in eq.4: (2 × F) – C = 2 + 1 + F + 1 which becomes 2×F – C = 4 + F In the above equation, add C to both sides, and subtract F from both sides: 2×F – C + C – F = 4 + F + C – F which becomes eq.4a) F = 4 + C
Hint #7
In eq.4a, substitute 4 – B for F (from eq.3a): 4 – B = 4 + C In the above equation, add B to both sides, and subtract 4 from both sides: 4 – B + B – 4 = 4 + C + B – 4 which becomes 0 = C + B Since C and B must be non-negative, the above equation makes: C = 0 and B = 0
Hint #8
Substitute 0 for C in eq.4a: F = 4 + 0 which makes F = 4
Solution
Substitute 4 for F in eq.2a: D = 1 + 4 which makes D = 5 and makes ABCDEF = 200514