Puzzle for September 21, 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.3, replace D with E + F (from eq.2): A = E + F + F which becomes eq.3a) A = E + 2×F
Hint #2
In eq.4, replace A with E + 2×F (from eq.3a): E - C = E + 2×F - E which becomes E - C = 2×F Add C to both sides of the above equation: E - C + C = 2×F + C which becomes eq.4a) E = 2×F + C
Hint #3
In eq.5, substitute 2×F + C for E (from eq.4a): C - F = 2×F + C - C which becomes C - F = 2×F Add F to both sides of the equation above: C - F + F = 2×F + F which makes C = 3×F
Hint #4
Substitute 3×F for C in eq.4a: E = 2×F + 3×F which makes E = 5×F
Hint #5
Substitute 5×F for E in eq.3a: A = 5×F + 2×F which makes A = 7×F
Hint #6
Substitute 5×F for E in eq.2: D = 5×F + F which makes D = 6×F
Hint #7
Substitute 3×F for C, and 6×F for D in eq.6: B + F = 3×F + 6×F - F which becomes B + F = 8×F Subtract F from each side of the above equation: B + F - F = 8×F - F which makes B = 7×F
Solution
Substitute 7×F for A and B, 3×F for C, 6×F for D, and 5×F for E in eq.1: 7×F + 7×F + 3×F + 6×F + E + 5×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 = B = 7×F = 7 × 1 = 7 C = 3×F = 3 × 1 = 3 D = 6×F = 6 × 1 = 6 E = 5×F = 5 × 1 = 5 and ABCDEF = 773651