Puzzle for August 1, 2022 ( )
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.5, replace B with A + F (from eq.4): A + A + F = D + F which becomes 2×A + F = D + F Subtract F from each side of the equation above: 2×A + F – F = D + F – F which makes 2×A = D
Hint #2
In eq.2, replace D with 2×A: E = A + 2×A which makes E = 3×A
Hint #3
In eq.6, substitute 2×A for D, and 3×A for E: 2×A + 3×A = F – 3×A which becomes 5×A = F – 3×A Add 3×A to both sides of the above equation: 5×A + 3×A = F – 3×A + 3×A which makes 8×A = F
Hint #4
Substitute 8×A for F in eq.4: B = A + 8×A which makes B = 9×A
Hint #5
Substitute 8×A for F in eq.3: 8×A = A + C Subtract A from both sides of the equation above: 8×A – A = A + C – A which makes 7×A = C
Solution
Substitute 9×A for B, 7×A for C, 2×A for D, 3×A for E, and 8×A for F in eq.1: A + 9×A + 7×A + 2×A + 3×A + 8×A = 30 which simplifies to 30×A = 30 Divide both sides of the above equation by 30: 30×A ÷ 30 = 30 ÷ 30 which means A = 1 making B = 9×A = 9 × 1 = 9 C = 7×A = 7 × 1 = 7 D = 2×A = 2 × 1 = 2 E = 3×A = 3 × 1 = 3 F = 8×A = 8 × 1 = 8 and ABCDEF = 197238