Puzzle for September 13, 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 C with E + F (from eq.2): A + F = E + F + E which becomes A + F = 2×E + F Subtract F from both sides of the above equation: A + F – F = 2×E + F – F which makes eq.6a) A = 2×E
Hint #2
In eq.5, replace C with E + F (from eq.2), and 2×E for A (from eq.6a): E + F + F = 2×E – F which becomes E + 2×F = 2×E – F In the above equation, subtract E from both sides, and add F to both sides: E + 2×F – E + F = 2×E – F – E + F which makes 3×F = E
Hint #3
In eq.6a, substitute (3×F) for E: A = 2×(3×F) which makes A = 6×F
Hint #4
Substitute 3×F for E in eq.2: C = 3×F + F which makes C = 4×F
Hint #5
Substitute 4×F for C in eq.3: B = 4×F + F which makes B = 5×F
Hint #6
Substitute 5×F for B, and 3×F for E in eq.4: D = 5×F + 3×F which makes D = 8×F
Solution
Substitute 6×F for A, 5×F for B, 4×F for C, 8×F for D, and 3×F for E in eq.1: 6×F + 5×F + 4×F + 8×F + 3×F + F = 27 which simplifies to 27×F = 27 Divide both sides of the above equation by 27: 27×F ÷ 27 = 27 ÷ 27 which means F = 1 making A = 6×F = 6 × 1 = 6 B = 5×F = 5 × 1 = 5 C = 4×F = 4 × 1 = 4 D = 8×F = 8 × 1 = 8 E = 3×F = 3 × 1 = 3 and ABCDEF = 654831