Puzzle for May 18, 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 B + D with A + C + E (from eq.5): A + C + E = A – C + E + F Subtract A and E from each side of the above equation: A + C + E – A – E = A – C + E + F – A – E which becomes C = –C + F Add C to both sides: C + C = –C + F + C which makes 2×C = F
Hint #2
In eq.3, replace F with 2×C: C + 2×C = A which makes 3×C = A
Hint #3
In eq.6, substitute A + D for E + F (from eq.4): B + D = A – C + A + D which becomes B + D = 2×A – C + D Subtract D from each side of the above equation: B + D – D = 2×A – C + D – D which makes eq.6a) B = 2×A – C
Hint #4
Substitute (3×C) for A in eq.6a: B = 2×(3×C) – C which becomes B = 6×C – C which makes B = 5×C
Hint #5
Substitute 5×C for B in eq.2: D = 5×C + C which makes D = 6×C
Hint #6
Substitute 2×C for F, 3×C for A, and 6×C for D in eq.4: E + 2×C = 3×C + 6×C which becomes E + 2×C = 9×C Subtract 2×C from each side of the equation above: E + 2×C – 2×C = 9×C – 2×C which makes E = 7×C
Solution
Substitute 3×C for A, 5×C for B, 6×C for D, 7×C for E, and 2×C for F in eq.1: 3×C + 5×C + C + 6×C + 7×C + 2×C = 24 which simplifies to 24×C = 24 Divide both sides of the above equation by 24: 24×C ÷ 24 = 24 ÷ 24 which means C = 1 making A = 3×C = 3 × 1 = 3 B = 5×C = 5 × 1 = 5 D = 6×C = 6 × 1 = 6 E = 7×C = 7 × 1 = 7 F = 2×C = 2 × 1 = 2 and ABCDEF = 351672