Puzzle for May 10, 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.4, replace A + B with C + E (from eq.2): E + F = C + E + C which becomes E + F = 2×C + E Subtract E from both sides of the above equation: E + F – E = 2×C + E – E which makes F = 2×C
Hint #2
In eq.6, replace F with 2×C: A = C + 2×C which makes A = 3×C
Hint #3
In eq.5, substitute 3×C for A, and 2×C for F: D = 3×C + C + 2×C which makes D = 6×C
Hint #4
Substitute 6×C for D in eq.3: B = C + 6×C which makes B = 7×C
Hint #5
Substitute 3×C for A, and 7×C for B in eq.2: C + E = 3×C + 7×C which becomes C + E = 10×C Subtract C from each side of the above equation: C + E – C = 10×C – C which makes E = 9×C
Solution
Substitute 3×C for A, 7×C for B, 6×C for D, 9×C for E, and 2×C for F in eq.1: 3×C + 7×C + C + 6×C + 9×C + 2×C = 28 which simplifies to 28×C = 28 Divide both sides of the above equation by 28: 28×C ÷ 28 = 28 ÷ 28 which means C = 1 making A = 3×C = 3 × 1 = 3 B = 7×C = 7 × 1 = 7 D = 6×C = 6 × 1 = 6 E = 9×C = 9 × 1 = 9 F = 2×C = 2 × 1 = 2 and ABCDEF = 371692