Puzzle for May 18, 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.3, replace F with A + D (from eq.2): D + A + D = A + B which becomes A + 2×D = A + B Subtract A from both sides of the above equation: A + 2×D – A = A + B – A which makes 2×D = B
Hint #2
In eq.4, replace B with 2×D, and F with A + D (from eq.2): A + E = 2×D + D + A + D which becomes A + E = 4×D + A Subtract A from both sides of the equation above: A + E – A = 4×D + A – A which makes E = 4×D
Hint #3
In eq.6, substitute 4×D for E: C = D + 4×D which makes C = 5×D
Hint #4
Substitute 2×D for B, and 5×D for C in eq.5: 2×D + 5×D = A which makes 7×D = A
Hint #5
Substitute 7×D for A in eq.2: F = 7×D + D which makes F = 8×D
Solution
Substitute 7×D for A, 2×D for B, 5×D for C, 4×D for E, and 8×D for F in eq.1: 7×D + 2×D + 5×D + D + 4×D + 8×D = 27 which simplifies to 27×D = 27 Divide both sides of the equation above by 27: 27×D ÷ 27 = 27 ÷ 27 which means D = 1 making A = 7×D = 7 × 1 = 7 B = 2×D = 2 × 1 = 2 C = 5×D = 5 × 1 = 5 E = 4×D = 4 × 1 = 4 F = 8×D = 8 × 1 = 8 and ABCDEF = 725148