Puzzle for December 12, 2022 ( )
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 D with E + F (from eq.4): F - E = E + F - F which becomes F - E = E Add E to both sides of the equation above: F - E + E = E + E which makes F = 2×E
Hint #2
In eq.4, substitute 2×E for F: D = E + 2×E which makes D = 3×E
Hint #3
Substitute 2×E for F, and 3×E for D in eq.5: E + 2×E = A - 3×E which becomes 3×E = A - 3×E Add 3×E to both sides of the above equation: 3×E + 3×E = A - 3×E + 3×E which makes 6×E = A
Hint #4
Substitute 6×E for A, and 2×E for F in eq.2: B = 6×E + 2×E which makes B = 8×E
Hint #5
Substitute 6×E for A, and 3×E for D in eq.3: C = 6×E + 3×E which makes C = 9×E
Solution
Substitute 6×E for A, 8×E for B, 9×E for C, 3×E for D, and 2×E for F in eq.1: 6×E + 8×E + 9×E + 3×E + E + 2×E = 29 which simplifies to 29×E = 29 Divide both sides of the above equation by 29: 29×E ÷ 29 = 29 ÷ 29 which means E = 1 making A = 6×E = 6 × 1 = 6 B = 8×E = 8 × 1 = 8 C = 9×E = 9 × 1 = 9 D = 3×E = 3 × 1 = 3 F = 2×E = 2 × 1 = 2 and ABCDEF = 689312