Puzzle for November 29, 2021 ( )
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.6, replace D + E with A + B (from eq.5): A + B + C = A + B + F Subtract A and B from each side of the equation above: A + B + C – A – B = A + B + F – A – B which makes C = F
Hint #2
In eq.4, replace F with C: C + C = E – C which becomes 2×C = E – C Add C to both sides of the above equation: 2×C + C = E – C + C which makes 3×C = E
Hint #3
In eq.3, substitute 3×C for E: B = C + 3×C which makes B = 4×C
Hint #4
Substitute 4×C for B in eq.2: D = 4×C + C which makes D = 5×C
Hint #5
Substitute 4×C for B, 5×C for D, and 3×C for E in eq.5: 5×C + 3×C = A + 4×C which becomes 8×C = A + 4×C Subtract 4×C from each side of the above equation: 8×C – 4×C = A + 4×C – 4×C which makes 4×C = A
Solution
Substitute 4×C for A and B, 5×C for D, 3×C for E, and C for F in eq.1: 4×C + 4×C + C + 5×C + 3×C + C = 18 which simplifies to 18×C = 18 Divide both sides of the above equation by 18: 18×C ÷ 18 = 18 ÷ 18 which means C = 1 making A = B = 4×C = 4×1 = 4 D = 5×C = 5×1 = 5 E = 3×C = 3×1 = 3 F = C = 1 and ABCDEF = 441531