Puzzle for May 16, 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.4, replace D with E + F (from eq.2): B + E = E + F + F which becomes B + E = E + 2×F Subtract E from both sides of the equation above: B + E – E = E + 2×F – E which makes B = 2×F
Hint #2
In eq.5, replace B with 2×F: E – F = 2×F + F which becomes E – F = 3×F Add F to both sides of the above equation: E – F + F = 3×F + F which makes E = 4×F
Hint #3
In eq.2, substitute 4×F for E: D = 4×F + F which makes D = 5×F
Hint #4
Substitute 5×F for D, and 4×F for E in eq.3: A = 5×F + 4×F – F which makes A = 8×F
Hint #5
Substitute 2×F for B, 4×F for E, and 8×F for A in eq.6: 2×F + 4×F – C = 8×F – 2×F + C which becomes 6×F – C = 6×F + C Subtract 6×F from each side of the above equation: 6×F – C – 6×F = 6×F + C – 6×F which makes –C = C which means 0 = C
Solution
Substitute 8×F for A, 2×F for B, 0 for C, 5×F for D, and 4×F for E in eq.1: 8×F + 2×F + 0 + 5×F + 4×F + F = 20 which simplifies to 20×F = 20 Divide both sides of the above equation by 20: 20×F ÷ 20 = 20 ÷ 20 which means F = 1 making A = 8×F = 8 × 1 = 8 B = 2×F = 2 × 1 = 2 D = 5×F = 5 × 1 = 5 E = 4×F = 4 × 1 = 4 and ABCDEF = 820541