Puzzle for November 13, 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 C with B + D (from eq.3): B + E = B + D - E In the equation above, subtract B from both sides, and add E to both sides: B + E - B + E = B + D - E - B + E which makes 2×E = D
Hint #2
In eq.4, replace D with 2×E: A = 2×E + E which makes A = 3×E
Hint #3
In eq.5, substitute 2×E for D, and 3×E for A: F - 2×E = 3×E + 2×E which becomes F - 2×E = 5×E Add 2×E to both sides of the above equation: F - 2×E + 2×E = 5×E + 2×E which makes F = 7×E
Hint #4
Substitute 7×E for F, and 3×E for A in eq.2: 7×E = 3×E + B Subtract 3×E from both sides of the equation above: 7×E - 3×E = 3×E + B - 3×E which makes 4×E = B
Hint #5
Substitute 4×E for B, and 2×E for D in eq.3: C = 4×E + 2×E which makes C = 6×E
Solution
Substitute 3×E for A, 4×E for B, 6×E for C, 2×E for D, and 7×E for F in eq.1: 3×E + 4×E + 6×E + 2×E + E + 7×E = 23 which simplifies to 23×E = 23 Divide both sides of the above equation by 23: 23×E ÷ 23 = 23 ÷ 23 which means E = 1 making A = 3×E = 3 × 1 = 3 B = 4×E = 4 × 1 = 4 C = 6×E = 6 × 1 = 6 D = 2×E = 2 × 1 = 2 F = 7×E = 7 × 1 = 7 and ABCDEF = 346217