Puzzle for January 20, 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 F with A + C (from eq.3): D = A + C – C + A which makes D = 2×A
Hint #2
In eq.2, replace D with 2×A: B = A + 2×A which makes B = 3×A
Hint #3
In eq.5, substitute 3×A for B: E = 3×A – A which makes E = 2×A
Hint #4
Substitute 2×A for E and D, 3×A for B in eq.6: F – 2×A = 3×A + 2×A Add 2×A to both sides of the above equation: F – 2×A + 2×A = 3×A + 2×A + 2×A which makes F = 7×A
Hint #5
Substitute 7×A for F in eq.3: 7×A = A + C Subtract A from both sides of the above equation: 7×A – A = A + C – A which makes 6×A = C
Solution
Substitute 3×A for B, 6×A for C, 2×A for D and E, and 7×A for F eq.1: A + 3×A + 6×A + 2×A + 2×A + 7×A = 21 which simplifies to 21×A = 21 Divide both sides of the equation above by 21: 21×A ÷ 21 = 21 ÷ 21 which means A = 1 making B = 3×A = 3 × 1 = 3 C = 6×A = 6 × 1 = 6 D = E = 2×A = 2 × 1 = 2 F = 7×A = 7 × 1 = 7 and ABCDEF = 136227