Puzzle for February 27, 2023 ( )
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 A with D + F (from eq.2): B + D = D + F + F which becomes B + D = D + 2×F Subtract D from each side of the equation above: B + D - D = D + 2×F - D which makes B = 2×F
Hint #2
In eq.6, replace B with 2×F: E - F = 2×F + F which becomes E - F = 3×F Add F to both sides of the above equation: E - F + F = 3×F + F which makes E = 4×F
Hint #3
In eq.4, substitute 4×F for E: C - 4×F = 4×F + F which becomes C - 4×F = 5×F Add 4×F to both sides of the equation above: C - 4×F - 4×F = 5×F + 4×F which makes C = 9×F
Hint #4
Substitute 2×F for B, and 4×F for E in eq.3: D - 2×F = 2×F + 4×F which becomes D - 2×F = 6×F Add 2×F to both sides of the above equation: D - 2×F + 2×F = 6×F + 2×F which makes D = 8×F
Hint #5
Substitute 8×F for D in eq.2: A = 8×F + F which makes A = 9×F
Solution
Substitute 9×F for A and C, 2×F for B, 8×F for D, and 4×F for E in eq.1: 9×F + 2×F + 9×F + 8×F + 4×F + F = 33 which simplifies to 33×F = 33 Divide both sides of the above equation by 33: 33×F ÷ 33 = 33 ÷ 33 which means F = 1 making A = C = 9×F = 9 × 1 = 9 B = 2×F = 2 × 1 = 2 D = 8×F = 8 × 1 = 8 E = 4×F = 4 × 1 = 4 and ABCDEF = 929841