Puzzle for July 28, 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.
* EF is a 2-digit number (not E×F).
Scratchpad
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Hint #1
Add D and F to both sides of eq.3: F - D + D + F = B - F + D + F which becomes eq.3a) 2×F = B + D Add F to both sides of eq.4: E + F + F = C + D - F + F which becomes eq.4a) E + 2×F = C + D
Hint #2
In eq.4a, replace 2×F with B + D (from eq.3a): E + B + D = C + D Subtract D from each side of the equation above: E + B + D - D = C + D - D which becomes eq.4b) E + B = C
Hint #3
In eq.5, replace C with E + B (from eq.4b): B - D + E = A - B + E + B + D - E which becomes B - D + E = A + D Add D to both sides of the equation above: B - D + E + D = A + D + D which becomes eq.5a) B + E = A + 2×D
Hint #4
In eq.5a, substitute A + D for B + E (from eq.2): A + D = A + 2×D Subtract A and D from both sides of the above equation: A + D - A - D = A + 2×D - A - D which makes 0 = D
Hint #5
Substitute 0 for D in eq.3a: 2×F = B + 0 which makes eq.3b) 2×F = B
Hint #6
Substitute 2×F for B, and 0 for D in eq.2: 2×F + E = A + 0 which becomes eq.2a) 2×F + E = A
Hint #7
eq.6 may be written as: eq.6a) 10×E + F = A + B + D
Hint #8
Substitute 2×F + E for A (from eq.2a), 2×F for B, and 0 for D into eq.6a: 10×E + F = 2×F + E + 2×F + 0 which becomes 10×E + F = 4×F + E Subtract F and E from both sides of the equation above: 10×E + F - F - E = 4×F + E - F - E which makes 9×E = 3×F Divide both sides by 3: 9×E ÷ 3 = 3×F ÷ 3 which makes 3×E = F
Hint #9
Substitute (3×E) for F in eq.3b: 2×(3×E) = B which makes 6×E = B
Hint #10
Substitute (3×E) for F in eq.2a: 2×(3×E) + E = A which becomes 6×E + E = A which makes 7×E = A
Hint #11
Substitute 6×E for B in eq.4b: E + 6×E = C which makes 7×E = C
Solution
Substitute 7×E for A and C, 6×E for B, 0 for D, and 3×E for F in eq.1: 7×E + 6×E + 7×E + 0 + E + 3×E = 24 which simplifies to 24×E = 24 Divide both sides of the above equation by 24: 24×E ÷ 24 = 24 ÷ 24 which means E = 1 making A = C = 7×E = 7 × 1 = 7 B = 6×E = 6 × 1 = 6 F = 3×E = 3 × 1 = 3 and ABCDEF = 767013