Puzzle for October 26, 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 E with C + D (from eq.4): C + C + D = A – C which becomes 2×C + D = A – C Add C to both sides of the equation above: 2×C + D + C = A – C + C which becomes eq.6a) 3×C + D = A
Hint #2
In eq.5, replace A with 3×C + D (from eq.6a): 3×C + D = B + D Subtract D from each side of the equation above: 3×C + D – D = B + D – D which makes 3×C = B
Hint #3
In eq.2, substitute 3×C for B: D = 3×C + C which makes D = 4×C
Hint #4
Substitute 4×C for D in eq.6a: 3×C + 4×C = A which makes 7×C = A
Hint #5
Substitute 4×C for D in eq.4: E = C + 4×C which makes E = 5×C
Hint #6
Substitute 4×C for D, and 5×C for E into eq.3: F = 4×C + 5×C which makes F = 9×C
Solution
Substitute 7×C for A, 3×C for B, 4×C for D, 5×C for E, and 9×C for F in eq.1: 7×C + 3×C + C + 4×C + 5×C + 9×C = 29 which simplifies to 29×C = 29 Divide both sides of the above equation by 29: 29×C ÷ 29 = 29 ÷ 29 which means C = 1 making A = 7×C = 7 × 1 = 7 B = 3×C = 3 × 1 = 3 D = 4×C = 4 × 1 = 4 E = 5×C = 5 × 1 = 5 F = 9×C = 9 × 1 = 9 and ABCDEF = 731459