Puzzle for October 2, 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 E with C + D (from eq.2), and A with B + C (from eq.3): C + C + D = B + C + B + D which becomes 2×C + D = 2×B + C + D Subtract C and D from both sides of the above equation: 2×C + D - C - D = 2×B + C + D - C - D which makes eq.6a) C = 2×B
Hint #2
In eq.5, replace C with 2×B: B + F = 2×B - F In the equation above, subtract B from both sides, and add F to both sides: B + F - B + F = 2×B - F - B + F which makes 2×F = B
Hint #3
In eq.6a, substitute (2×F) for B: C = 2×(2×F) which makes C = 4×F
Hint #4
Substitute 4×F for C in eq.4: D = 4×F + F which makes D = 5×F
Hint #5
Substitute 4×F for C, and 5×F for D in eq.2: E = 4×F + 5×F which makes E = 9×F
Hint #6
Substitute 2×F for B, and 4×F for C in eq.3: A = 2×F + 4×F which makes A = 6×F
Solution
Substitute 6×F for A, 2×F for B, 4×F for C, 5×F for D, and 9×F for E in eq.1: 6×F + 2×F + 4×F + 5×F + 9×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 = 6×F = 6×F = 6 B = 2×F = 2×F = 2 C = 4×F = 4×F = 4 D = 5×F = 5×F = 5 E = 9×F = 9×F = 9 and ABCDEF = 624591