Puzzle for September 6, 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.3, replace C with A + D (from eq.4): A + A + D = B + D which becomes 2×A + D = B + D Subtract D from each side of the above equation: 2×A + D – D = B + D – D which makes 2×A = B
Hint #2
In eq.2, replace B with 2×A: A + 2×A = E – 2×A which becomes 3×A = E – 2×A Add 2×A to both sides of the above equation: 3×A + 2×A = E – 2×A + 2×A which makes 5×A = E
Hint #3
In eq.6, substitute 5×A for E: A + F = 5×A Subtract A from each side of the above equation: A + F – A = 5×A – A which makes F = 4×A
Hint #4
Substitute 5×A for E, and 2×A for B in eq.5: A + 5×A = C – 2×A which becomes 6×A = C – 2×A Add 2×A to both sides of the above equation: 6×A + 2×A = C – 2×A + 2×A which makes 8×A = C
Hint #5
Substitute 8×A for C in eq.4: A + D = 8×A Subtract A from each side of the equation above: A + D – A = 8×A – A which makes D = 7×A
Solution
Substitute 2×A for B, 8×A for C, 7×A for D, 5×A for E, and 4×A for F in eq.1: A + 2×A + 8×A + 7×A + 5×A + 4×A = 27 which simplifies to 27×A = 27 Divide both sides of the above equation by 27: 27×A ÷ 27 = 27 ÷ 27 which means A = 1 making B = 2×A = 2 × 1 = 2 C = 8×A = 8 × 1 = 8 D = 7×A = 7 × 1 = 7 E = 5×A = 5 × 1 = 5 F = 4×A = 4 × 1 = 4 and ABCDEF = 128754