Puzzle for August 14, 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 F to both sides of eq.4: D - F + F = E + F + F which becomes eq.4a) D = E + 2×F In eq.6, replace D with E + 2×F (from eq.4a): E - F = E + 2×F - E which becomes E - F = 2×F Add F to both sides of the equation above: E - F + F = 2×F + F which makes E = 3×F
Hint #2
In eq.4a, substitute 3×F for E: D = 3×F + 2×F which makes D = 5×F
Hint #3
In eq.3, substitute 3×F for E: A = 3×F + F which makes A = 4×F
Hint #4
Substitute 4×F for A, and 3×F for E in eq.2: C = 4×F + 3×F which makes C = 7×F
Hint #5
Substitute 7×F for C, and 3×F for E in eq.5: B + F = 7×F + 3×F which becomes B + F = 10×F Subtract F from each side of the equation above: B + F - F = 10×F - F which makes B = 9×F
Solution
Substitute 4×F for A, 9×F for B, 7×F for C, 5×F for D, and 3×F for E in eq.1: 4×F + 9×F + 7×F + 5×F + 3×F + F = 29 which simplifies to 29×F = 29 Divide both sides of the above equation by 29: 29×F ÷ 29 = 29 ÷ 29 which means F = 1 making A = 4×F = 4 × 1 = 4 B = 9×F = 9 × 1 = 9 C = 7×F = 7 × 1 = 7 D = 5×F = 5 × 1 = 5 E = 3×F = 3 × 1 = 3 and ABCDEF = 497531