Puzzle for September 11, 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 C with A + F (from eq.4): A + E = A + F - E In the above equation, subtract A from both sides, and add E to both sides: A + E - A + E = A + F - E - A + E which makes 2×E = F
Hint #2
In eq.6, replace F with 2×E: E + 2×E = C - 2×E which becomes 3×E = C - 2×E Add 2×E to both sides of the above equation: 3×E + 2×E = C - 2×E + 2×E which makes 5×E = C
Hint #3
In eq.5, substitute 5×E for C: A + E = 5×E - E which becomes A + E = 4×E Subtract E from both sides of the equation above: A + E - E = 4×E - E which makes A = 3×E
Hint #4
Substitute 3×E for A, and 5×E for C in eq.2: D = 3×E + 5×E which makes D = 8×E
Hint #5
Substitute 8×E for D in eq.3: B = 8×E + E which makes B = 9×E
Solution
Substitute 3×E for A, 9×E for B, 5×E for C, 8×E for D, and 2×E for F in eq.1: 3×E + 9×E + 5×E + 8×E + E + 2×E = 28 which simplifies to 28×E = 28 Divide both sides of the above equation by 28: 28×E ÷ 28 = 28 ÷ 28 which means E = 1 making A = 3×E = 3 × 1 = 3 B = 9×E = 9 × 1 = 9 C = 5×E = 5 × 1 = 5 D = 8×E = 8 × 1 = 8 F = 2×E = 2 × 1 = 2 and ABCDEF = 395812