Puzzle for June 1, 2020 ( )
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 C + E with A (from eq.6): A + D = A + F Subtract A from both sides of the above equation: A + D – A = A + F – A which makes D = F
Hint #2
In eq.3, replace D with F, and A + B with F (from eq.2): F + E = F + C Subtract F from each side of the above equation: F + E – F = F + C – F which makes E = C
Hint #3
In eq.6, substitute C for E: C + C = A which makes 2×C = A
Hint #4
Substitute 2×C for A, and C for E in eq.5: B = 2×C + C which makes B = 3×C
Hint #5
Substitute 2×C for A, and 3×C for B in eq.2: F = 2×C + 3×C which makes F = 5×C and also makes D = F = 5×C
Solution
Substitute 2×C for A, 3×C for B, 5×C for D and F, and C for E in eq.1: 2×C + 3×C + C + 5×C + C + 5×C = 17 which simplifies to 17×C = 17 Divide both sides of the above equation by 17: 17×C ÷ 17 = 17 ÷ 17 which means C = 1 making A = 2×C = 2 B = 3×C = 3 D = F = 5×C = 5 E = C = 1 and ABCDEF = 231515