Puzzle for August 30, 2021 ( )
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 D with E + F (from eq.2), and A with D + E (from eq.3): E + F + F = D + E – F which becomes E + 2×F = D + E – F In the above equation, subtract E from both sides, and add F to both sides: E + 2×F – E + F = D + E – F – E + F which makes 3×F = D
Hint #2
In eq.2, replace D with 3×F: E + F = 3×F Subtract F from each side of the equation above: E + F – F = 3×F – F which makes E = 2×F
Hint #3
In eq.6, substitute 3×F for D: 3×F + F = A – F which makes 4×F = A – F Add F to both sides of the equation above: 4×F + F = A – F + F which makes 5×F = A
Hint #4
Substitute 5×F for A in eq.4: 5×F + F = C which makes 6×F = C
Hint #5
Substitute 5×F for A, and 2×F for E in eq.5: B = 5×F + 2×F + F which makes B = 8×F
Solution
Substitute 5×F for A, 8×F for B, 6×F for C, 3×F for D, and 2×F for E in eq.1: 5×F + 8×F + 6×F + 3×F + 2×F + F = 25 which simplifies to 25×F = 25 Divide both sides of the above equation by 25: 25×F ÷ 25 = 25 ÷ 25 which means F = 1 making A = 5×F = 5 × 1 = 5 B = 8×F = 8 × 1 = 8 C = 6×F = 6 × 1 = 6 D = 3×F = 3 × 1 = 3 E = 2×F = 2 × 1 = 2 and ABCDEF = 586321