Puzzle for December 7, 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.3, replace B with A + F (from eq.2): A + A + F = E + F which becomes 2×A + F = E + F Subtract F from each side of the equation above: 2×A + F - F = E + F - F which makes 2×A = E
Hint #2
In eq.6, replace B with A + F (from eq.2), and E with 2×A: A + F + C - F = 2×A + F which becomes A + C = 2×A + F Subtract A from both sides of the above equation: A + C - A = 2×A + F - A which becomes eq.6a) C = A + F
Hint #3
In eq.6a, substitute B for A + F (from eq.2): C = B
Hint #4
In eq.5, replace C with A + F (from eq.6a), and E with 2×A: A + F + 2×A = D + F which becomes 3×A + F = D + F Subtract F from each side of the above equation: 3×A + F - F = D + F - F which makes 3×A = D
Hint #5
Substitute 3×A for D, and 2×A for E in eq.4: F - 3×A = 3×A + 2×A which becomes F - 3×A = 5×A Add 3×A to both sides of the above equation: F - 3×A + 3×A = 5×A + 3×A which makes F = 8×A
Hint #6
Substitute 8×A for F in eq.6a: A + 8×A = C which makes 9×A = C and also makes 9×A = C = B
Solution
Substitute 9×A for B and C, 3×A for D, 2×A for E, and 8×A for F in eq.1: A + 9×A + 9×A + 3×A + 2×A + 8×A = 32 which simplifies to 32×A = 32 Divide both sides of the above equation by 32: 32×A ÷ 32 = 32 ÷ 32 which means A = 1 making B = C = 9×A = 9 × 1 = 9 D = 3×A = 3 × 1 = 3 E = 2×A = 2 × 1 = 2 F = 8×A = 8 × 1 = 8 and ABCDEF = 199328