Puzzle for September 20, 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.4, replace C with A – D + F (from eq.3): D = A – D + F – F which becomes D = A – D Add D to both sides of the above equation: D + D = A – D + D which makes 2×D = A
Hint #2
In eq.5, replace A with 2×D: E = 2×D + D which makes E = 3×D
Hint #3
In eq.6, substitute 2×D for A, and 3×D for E: F = 2×D + 3×D which makes F = 5×D
Hint #4
Substitute 5×D for F in eq.4: D = C – 5×D Add 5×D to both sides of the equation above: D + 5×D = C – 5×D + 5×D which makes 6×D = C
Hint #5
Substitute 6×D for C, and 3×D for E in eq.2: B = 6×D + 3×D which makes B = 9×D
Solution
Substitute 2×D for A, 9×D for B, 6×D for C, 3×D for E, and 5×D for F in eq.1: 2×D + 9×D + 6×D + D + 3×D + 5×D = 26 which simplifies to 26×D = 26 Divide both sides of the above equation by 26: 26×D ÷ 26 = 26 ÷ 26 which means D = 1 making A = 2×D = 2 × 1 = 2 B = 9×D = 9 × 1 = 9 C = 6×D = 6 × 1 = 6 E = 3×D = 3 × 1 = 3 F = 5×D = 5 × 1 = 5 and ABCDEF = 296135