Puzzle for July 15, 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 non-negative integer.
Scratchpad
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Hint #1
In eq.5, replace C with A + F (from eq.4): D + F = A + F - D In the above equation, subtract F from both sides, and add D to both sides: D + F - F + D = A + F - D - F + D which simplifies to 2×D = A
Hint #2
In eq.2, replace A with 2×D: B = 2×D + D which makes B = 3×D
Hint #3
In eq.6, replace C with A + F (from eq.4), and D + E with F (from eq.3): A + A + F = F + F which becomes 2×A + F = 2×F Subtract F from both sides of the above equation: 2×A + F - F = 2×F - F which makes 2×A = F Replace A with (2×D): 2×(2×D) = F which makes 4×D = F
Hint #4
In eq.4, substitute 2×D for A, and 4×D for F: C = 2×D + 4×D which makes C = 6×D
Hint #5
Substitute 4×D for F in eq.3: 4×D = D + E Subtract D from both sides of the above equation: 4×D - D = D + E - D which makes 3×D = E
Solution
Substitute 2×D for A, 3×D for B and E, 6×D for C, and 4×D for F in eq.1: 2×D + 3×D + 6×D + D + 3×D + 4×D = 19 which simplifies to 19×D = 19 Divide both sides of the equation above by 19: 19×D ÷ 19 = 19 ÷ 19 which means D = 1 making A = 2×D = 2 × 1 = 1 B = E = 3×D = 3 × 1 = 3 C = 6×D = 6 × 1 = 6 F = 4×D = 4 × 1 = 4 and ABCDEF = 236134