Puzzle for May 6, 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.
Scratchpad
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Hint #1
Add A to both sides of eq.5: B - A + A = A + F + A which becomes eq.5a) B = 2×A + F In eq.3, replace B with 2×A + F: A + 2×A + F = D + F which becomes 3×A + F = D + F Subtract F from both sides of the above equation: 3×A + F - F = D + F - F which makes 3×A = D
Hint #2
In eq.6, replace D with 3×A: 3×A = C × A Divide both sides of the above equation by A: 3×A ÷ A = C × A ÷ A which makes 3 = C
Hint #3
Substitute 3 for C, and 3×A for D in eq.2: F = 3 + A + 3×A which means eq.2a) F = 3 + 4×A
Hint #4
Substitute 3 + 4×A for F (from eq.2a) in eq.5a: B = 2×A + 3 + 4×A which means eq.5b) B = 3 + 6×A
Hint #5
Substitute 3 for C, 3×A for D, and 3 + 6×A for B (from eq.5b) in eq.4: 3 + 3×A + E = 3 + 6×A Subtract both 3 and 3×A from each side of the above equation: 3 + 3×A + E - 3 - 3×A = 3 + 6×A - 3 - 3×A which makes E = 3×A
Solution
Substitute 3 + 6×A for B (from eq.5b), 3 for C, 3×A for D and E, and 3 + 4×A for F (from eq.2a) in eq.1: A + 3 + 6×A + 3 + 3×A + 3×A + 3 + 4×A = 26 which becomes 17×A + 9 = 26 Subtract 9 from both sides of the equation above: 17×A + 9 - 9 = 26 - 9 which simplifies to 17×A = 17 Divide both sides by 17: 17×A ÷ 17 = 17 ÷ 17 which means A = 1 making B = 3 + 6×A = 3 + 6×1 = 3 + 6 = 9 (from eq.5b) D = E = 3×A = 3×1 = 3 F = 3 + 4×A = 3 + 4×1 = 3 + 4 = 7 (from eq.2a) and ABCDEF = 193337