Puzzle for May 3, 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.6, replace C with A + D (from eq.2): A + D + D = A + B which becomes A + 2×D = A + B Subtract A from each side 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: F = 2×D + D which makes F = 3×D
Hint #3
In eq.5, substitute 2×D for B, and 3×D for F: 2×D + 3×D = A which makes 5×D = A
Hint #4
Substitute 5×D for A in eq.2: C = 5×D + D which makes C = 6×D
Hint #5
Substitute 2×D for B, and 6×D for C in eq.3: E = 2×D + 6×D which makes E = 8×D
Solution
Substitute 5×D for A, 2×D for B, 6×D for C, 8×D for E, and 3×D for F in eq.1: 5×D + 2×D + 6×D + D + 8×D + 3×D = 25 which simplifies to 25×D = 25 Divide both sides of the equation above by 25: 25×D ÷ 25 = 25 ÷ 25 which means D = 1 making A = 5×D = 5 × 1 = 5 B = 2×D = 2 × 1 = 2 C = 6×D = 6 × 1 = 6 E = 8×D = 8 × 1 = 8 F = 3×D = 3 × 1 = 3 and ABCDEF = 526183