Puzzle for January 18, 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.3, replace C with A + E (from eq.2): D + E = A + E – D In the equation above, subtract E from both sides, and add D to both sides: D + E – E + D = A + E – D – E + D which makes 2×D = A
Hint #2
In eq.5, replace A with 2×D: E – D = 2×D + D which becomes E – D = 3×D Add D to both sides of the above equation: E – D + D = 3×D + D which makes E = 4×D
Hint #3
In eq.2, substitute 2×D for A, and 4×D for E: C = 2×D + 4×D which makes C = 6×D
Hint #4
Substitute 6×D for C, 4×D for E, and 2×D for A in eq.4: B + 6×D – D – 4×D = 2×D + D + 4×D which becomes B + D = 7×D Subtract D from each side of the equation above: B + D – D = 7×D – D which makes B = 6×D
Hint #5
Substitute 6×D for B and C in eq.6: F = 6×D – 6×D which makes F = 0
Solution
Substitute 2×D for A, 6×D for B and C, 4×D for E, and 0 for F in eq.1: 2×D + 6×D + 6×D + D + 4×D + 0 = 19 which simplifies to 19×D = 19 Divide both sides of the equation above by 19: 19×D ÷ 19 = 19 ÷ 19 which means D = 1 making A = 2×D = 2 × 1 = 2 B = C = 6×D = 6 × 1 = 6 E = 4×D = 4 × 1 = 4 and ABCDEF = 266140