Puzzle for October 3, 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.6, replace E with D + F (from eq.2): A + D = B + D + F + F which becomes A + D = B + D + 2×F Subtract D from each side of the equation above: A + D - D = B + D + 2×F - D which becomes eq.6a) A = B + 2×F
Hint #2
In eq.3, replace A with B + 2×F (from eq.6a): B + F = B + 2×F - B which becomes B + F = 2×F Subtract F from both sides of the equation above: B + F - F = 2×F - F which makes B = F
Hint #3
In eq.6a, substitute B for F: A = B + 2×B which makes A = 3×B
Hint #4
Substitute 3×B for A, and B for F in eq.4: D - 3×B = 3×B + B + B which becomes D - 3×B = 5×B Add 3×B to both sides of the above equation: D - 3×B + 3×B = 5×B + 3×B which makes D = 8×B
Hint #5
Substitute 8×B for D, and B for F in eq.2: E = 8×B + B which makes E = 9×B
Hint #6
Substitute 9×B for E, and 3×B for A in eq.5: 9×B = 3×B + C Subtract 3×B from each side of the equation above: 9×B - 3×B = 3×B + C - 3×B which makes 6×B = C
Solution
Substitute 3×B for A, 6×B for C, 8×B for D, 9×B for E, and B for F in eq.1: 3×B + B + 6×B + 8×B + 9×B + B = 28 which simplifies to 28×B = 28 Divide both sides of the above equation by 28: 28×B ÷ 28 = 28 ÷ 28 which means B = 1 making A = 3×B = 3 × 1 = 3 C = 6×B = 6 × 1 = 6 D = 8×B = 8 × 1 = 8 E = 9×B = 9 × 1 = 9 F = B = 1 and ABCDEF = 316891