Puzzle for February 11, 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 E to both sides of eq.3: C - E + E = E + E which becomes C = 2×E
Hint #2
In eq.2, replace C with 2×E: 2×E + D = A + E Subtract E from both sides of the above equation: 2×E + D - E = A + E - E which becomes eq.2a) E + D = A In eq.4, replace A with E + D: E + D - E = F which makes D = F
Hint #3
In eq.5, substitute D for F: D - D = A - B which means 0 = A - B Add B to each side of the above equation: 0 + B = A - B + B which makes B = A
Hint #4
In eq.6, replace B with A: A = A × E Divide both sides of the above equation by A: A ÷ A = A × E ÷ A which makes 1 = E making C = 2 × 1 = 2
Hint #5
Substitute 1 for E in eq.2a: 1 + D = A
Solution
Substitute 1 + D for both A and B, 2 for C, 1 for E, and D for F in eq.1: 1 + D + 1 + D + 2 + D + 1 + D = 33 which simplifies to 4×D + 5 = 33 Subtract 5 from both sides of the above equation: 4×D + 5 - 5 = 33 - 5 which becomes 4×D = 28 Divide both sides by 4: 4×D ÷ 4 = 28÷ 4 which makes D = 7 making A = B = 1 + D = 1 + 7 = 8 D = F = 7 and ABCDEF = 882717