Puzzle for January 30, 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 B with D + E (from eq.2): D + E + D = E + F which becomes 2×D + E = E + F Subtract E from each side of the above equation: 2×D + E - E = E + F - E which makes 2×D = F
Hint #2
In eq.4, substitute 2×D for F: C = D + 2×D which makes C = 3×D
Hint #3
In eq.3, substitute 3×D for C, and 2×D for F: A = 3×D + 2×D which makes A = 5×D
Hint #4
Substitute 5×D for A, and 2×D for F in eq.5: E = 5×D + D + 2×D which makes E = 8×D
Hint #5
Substitute 8×D for E in eq.2: B = D + 8×D which makes B = 9×D
Solution
Substitute 5×D for A, 9×D for B, 3×D for C, 8×D for E, and 2×D for F in eq.1: 5×D + 9×D + 3×D + D + 8×D + 2×D = 28 which simplifies to 28×D = 28 Divide both sides of the above equation by 28: 28×D ÷ 28 = 28 ÷ 28 which means D = 1 making A = 5×D = 5 × 1 = 5 B = 9×D = 9 × 1 = 9 C = 3×D = 3 × 1 = 3 E = 8×D = 8 × 1 = 8 F = 2×D = 2 × 1 = 2 and ABCDEF = 593182