Puzzle for August 31, 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
Add D to both sides of eq.6: A - D + D = E + D which becomes eq.6a) A = E + D
Hint #2
In eq.2, replace A with E + D (from eq.6a): C - D = E + D - C In the above equation, subtract D from both sides, and add C to both sides: C - D - D + C = E + D - C - D + C which becomes eq.2a) 2×C - 2×D = E
Hint #3
In eq.4, substitute (2×C - 2×D) for E (from eq.2a): (2×C - 2×D) - D = C - (2×C - 2×D) which becomes 2×C - 3×D = C - 2×C + 2×D which becomes 2×C - 3×D = -C + 2×D Add 3×D and C to both sides of the above equation: 2×C - 3×D + 3×D + C = -C + 2×D + 3×D + C which makes eq.4a) 3×C = 5×D
Hint #4
Add D to both sides of eq.5: F - D + D = D - C + D which becomes F = 2×D - C Multiply both sides of the above equation by 3: 3×F = 3×(2×D - C) which becomes eq.5a) 3×F = 6×D - 3×C
Hint #5
In eq.5a, replace 3×C with 5×D: 3×F = 6×D - 5×D which makes 3×F = D
Hint #6
In eq.4a, substitute (3×F) for D: 3×C = 5×(3×F) which makes 3×C = 15×F Divide both sides of the above equation by 3: 3×C ÷ 3 = 15×F ÷ 3 which makes C = 5×F
Hint #7
Substitute 5×F for C, and 3×F for D in eq.2a: 2×(5×F) - 2×(3×F) = E which becomes 10×F - 6×F = E which makes 4×F = E
Hint #8
Substitute 4×F for E in eq.3: B - 4×F = 4×F - F which becomes B - 4×F = 3×F Add 4×F to both sides of the equation above: B - 4×F + 4×F = 3×F + 4×F which makes B = 7×F
Hint #9
Substitute 4×F for E, and 3×F for D in eq.6a: A = 4×F + 3×F which makes A = 7×F
Solution
Substitute 7×F for A and B, 5×F for C, 3×F for D, and 4×F for E in eq.1: 7×F + 7×F + 5×F + 3×F + 4×F + F = 27 which simplifies to 27×F = 27 Divide both sides of the above equation by 27: 27×F ÷ 27 = 27 ÷ 27 which means F = 1 making A = B = 7×F = 7 × 1 = 7 C = 5×F = 5 × 1 = 5 D = 3×F = 3 × 1 = 3 E = 4×F = 4 × 1 = 4 and ABCDEF = 775341