Puzzle for September 18, 2023 ( )
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 E with B + C (from eq.2): B + C - D = C + D In the equation above, subtract C from both sides, and add D to both sides: B + C - D - C + D = C + D - C + D which makes B = 2×D
Hint #2
In eq.3, replace B with 2×D: A = 2×D + D which makes A = 3×D
Hint #3
In eq.6, replace B with 2×D: F - 2×D = 2×D + D which becomes F - 2×D = 3×D Add 2×D to both sides of the above equation: F - 2×D + 2×D = 3×D + 2×D which makes F = 5×D
Hint #4
In eq.4, substitute 5×D for F: C = D + 5×D which makes C = 6×D
Hint #5
Substitute 2×D for B, and 6×D for C in eq.2: E = 2×D + 6×D which makes E = 8×D
Solution
Substitute 3×D for A, 2×D for B, 6×D for C, 8×D for E, and 5×D for F in eq.1: 3×D + 2×D + 6×D + D + 8×D + 5×D = 25 which simplifies to 25×D = 25 Divide both sides of the above equation by 25: 25×D ÷ 25 = 25 ÷ 25 which means D = 1 making A = 3×D = 3 × 1 = 3 B = 2×D = 2 × 1 = 2 C = 6×D = 6 × 1 = 6 E = 8×D = 8 × 1 = 8 F = 5×D = 5 × 1 = 5 and ABCDEF = 326185