Puzzle for December 2, 2019 ( )
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.2, replace A with C + D (from eq.3): D + F = C + D + C which becomes D + F = 2×C + D Subtract D from both sides of the equation above: D + F – D = 2×C + D – D which makes F = 2×C
Hint #2
In eq.4, replace F with 2×C: 2×C = B + C Subtract C from each side of the above equation: 2×C – C = B + C – C which makes C = B
Hint #3
In eq.6, substitute C + D for A (from eq.3), and 2×C for F: C + D + C + 2×C = D + E which becomes 4×C + D = D + E Subtract D from both sides of the above equation: 4×C + D – D = D + E – D which makes 4×C = E
Hint #4
Substitute 4×C for E, and 2×C for F in eq.5: 4×C + 2×C = D – 2×C Add 2×C to each side of the equation above: 4×C + 2×C + 2×C = D – 2×C + 2×C which makes 8×C = D
Hint #5
Substitute 8×C for D in eq.3: C + 8×C = A which makes 9×C = A
Solution
Substitute 9×C for A, C for B, 8×C for D, 4×C for E, and 2×C for F in eq.1: 9×C + C + C + 8×C + 4×C + 2×C = 25 which simplifies to 25×C = 25 Divide both sides of the equation above by 25: 25×C ÷ 25 = 25 ÷ 25 which means C = 1 making A = 9×C = 9×1 = 9 B = C = 1 D = 8×C = 8×1 = 8 E = 4×C = 4×1 = 4 F = 2×C = 2×1 = 2 and ABCDEF = 911842