Puzzle for December 19, 2018 ( )
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
In eq.3, replace D + E with A + B (from eq.2): A + C = A + B + F Subtract A from each side of the above equation: A + C - A = A + B + F - A which becomes eq.3a) C = B + F In eq.5, replace B + F with C: C = A
Hint #2
In eq.1, replace C with B + F (from eq.3a): B + F + F = B + D Subtract B from each side: B + F + F - B = B + D - B which means 2×F = D
Hint #3
In eq.6, replace C with A: E - B = A ÷ A which becomes E - B = 1 Add B to each side of the above equation: E - B + B = 1 + B which makes eq.6a) E = 1 + B
Hint #4
Substitute 1 + B for E (from eq.6a) in eq.2: D + 1 + B = A + B Subtract B from each side of the above equation: D + 1 + B - B = A + B - B which makes D + 1 = A Substitute 2×F for D: eq.2a) 2×F + 1 = A
Hint #5
In eq.5, replace A with 2×F + 1 (from eq.2a): B + F = 2×F + 1 Subtract F from each side of the above equation: B + F - F = 2×F + 1 - F which makes eq.5a) B = F + 1
Hint #6
Substitute F + 1 for B (from eq.5a) in eq.6a: E = 1 + F + 1 which makes eq.6b) E = F + 2
Solution
Substitute F + 1 for B (from eq.5a), 2×F for D, and (F + 2) for E (from eq.6b) in eq.4: F + 1 - F = 2×F - (F + 2) which becomes 1 = 2×F - F - 2 which becomes 1 = F - 2 Add 2 to both sides of the above equation: 1 + 2 = F - 2 + 2 which makes 3 = F making A = C = 2×F + 1 = 2×3 + 1 = 6 + 1 = 7 (from eq.2a) B = F + 1 = 3 + 1 = 4 (from eq.5a) D = 2×F = 2×3 = 6 E = F + 2 = 3 + 2 = 5 (from eq.6b) and ABCDEF = 747653