Puzzle for April 17, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit positive integer.
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Hint #1
Add A + D to both sides of eq.3: E - A + A + D = A - D + A + D which becomes E + D = 2×A which may be written as eq.3a) D + E = 2×A
Hint #2
In eq.6, replace D + E with 2×A (from eq.3a): A × D = 2×A Divide both sides of the above equation by A: A × D ÷ A = 2×A ÷ A which makes D = 2
Hint #3
In eq.2, replace D with 2: eq.2a) A + 2 = F
Hint #4
In eq.3a, substitute 2 for D: 2 + E = 2×A Subtract 2 from each side of the above equation: 2 + E - 2 = 2×A - 2 which makes eq.3b) E = 2×A - 2
Hint #5
Substitute 2 for D, and (2×A - 2) for E (from eq.3b) in eq.4: 2 - C = A - 2 - (2×A - 2) which is equivalent to 2 - C = A - 2 - 2×A + 2 which becomes 2 - C = -A Add A + C to both sides: 2 - C + A + C = -A + A + C which simplifies to 2 + A = C which may be written as eq.4a) A + 2 = C
Hint #6
Substitute 2 for D, 2×A - 2 for E (from eq.3b), A + 2 for C (from eq.4a) and for F (from eq.2a) in eq.5: B + 2 + 2×A - 2 = A + A + 2 - 2 + A + 2 which simplifies to B + 2×A = 3×A + 2 Subtract 2×A from both sides: B + 2×A - 2×A = 3×A + 2 - 2×A which becomes eq.5a) B = A + 2
Solution
Substitute A + 2 for B and C and F (from eq.5a, eq.4a, and eq.2a), 2 for D, and 2×A - 2 for E (from eq.3b) in eq.1: A + A + 2 + A + 2 + 2 + 2×A - 2 + A + 2 = 36 which simplifies to 6×A + 6 = 36 Subtract 6 from each side: 6×A + 6 - 6 = 36 - 6 which makes 6×A = 30 Divide both sides by 6: 6×A ÷ 6 = 30 ÷ 6 which means A = 5 making B = C = F = A + 2 = 5 + 2 = 7 (from eq.5a, eq.4a, and eq.2a) E = 2×A - 2 = 2×5 - 2 = 10 - 2 = 8 (from eq.3b) and ABCDEF = 577287