Puzzle for October 6, 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.5, replace B with C + E (from eq.3): C + F = C + E - F In the above equation, subtract C from both sides, and add F to both sides: C + F - C + F = C + E - F - C + F which makes 2×F = E
Hint #2
In eq.6, substitute (A + F) for D (from eq.2): F - A = A - C - (A + F) which becomes F - A = A - C - A - F which becomes F - A = -C - F Add A, C, and F to both sides of the equation above: F - A + A + C + F = -C - F + A + C + F which simplifies to eq.6a) 2×F + C = A
Hint #3
In eq.6a, replace 2×F with E: E + C = A which may be written as eq.6b) C + E = A
Hint #4
In eq.6b, substitute B for C + E (from eq.3): B = A
Hint #5
Substitute 2×F for E, and (A + F) for D (from eq.2) in eq.4: 2×F - C = C - (A + F) which becomes 2×F - C = C - A - F Add C, A, and F to both sides of the above equation: 2×F - C + C + A + F = C - A - F + C + A + F which becomes eq.4a) 3×F + A = 2×C
Hint #6
Substitute 2×F + C for A (from eq.6a) into eq.4a: 3×F + 2×F + C = 2×C which becomes 5×F + C = 2×C Subtract C from each side of the equation above: 5×F + C - C = 2×C - C which makes 5×F = C
Hint #7
Substitute 5×F for C in eq.6a: 2×F + 5×F = A which makes 7×F = A and also makes 7×F = A = B
Hint #8
Substitute 7×F for A in eq.2: D = 7×F + F which makes D = 8×F
Solution
Substitute 7×F for A and B, 5×F for C, 8×F for D, and 2×F for E in eq.1: 7×F + 7×F + 5×F + 8×F + 2×F + F = 30 which simplifies to 30×F = 30 Divide both sides of the above equation by 30: 30×F ÷ 30 = 30 ÷ 30 which means F = 1 making A = B = 7×F = 7 × 1 = 7 C = 5×F = 5 × 1 = 5 D = 8×F = 8 × 1 = 8 E = 2×F = 2 × 1 = 2 and ABCDEF = 775821