Puzzle for December 13, 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.5, replace C with A + D + E (from eq.6): A + D + E – E = A + E which becomes A + D = A + E Subtract A from each side of the equation above: A + D – A = A + E – A which makes D = E
Hint #2
In eq.2, replace E with D: D + D = F which makes 2×D = F
Hint #3
In eq.3, substitute 2×D for F: D + 2×D = A which makes 3×D = A
Hint #4
Substitute D for E, and 3×D for A in eq.5: C – D = 3×D + D which becomes C – D = 4×D Add D to both sides of the above equation: C – D + D = 4×D + D which makes C = 5×D
Hint #5
Substitute 5×D for C in eq.4: 5×D + D = B which makes 6×D = B
Solution
Substitute 3×D for A, 6×D for B, 5×D for C, D for E, and 2×D for F in eq.1: 3×D + 6×D + 5×D + D + D + 2×D = 18 which simplifies to 18×D = 18 Divide both sides of the above equation by 18: 18×D ÷ 18 = 18 ÷ 18 which means D = 1 making A = 3×D = 3 × 1 = 3 B = 6×D = 6 × 1 = 6 C = 5×D = 5 × 1 = 5 E = D = 1 F = 2×D = 2 × 1 = 2 and ABCDEF = 365112