Puzzle for February 14, 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.5, replace C with E + F (from eq.4): B + E = E + F + F which becomes B + E = E + 2×F Subtract E from each side of the equation above: B + E – E = E + 2×F – E which makes B = 2×F
Hint #2
In eq.6, replace B with 2×F: D – 2×F = 2×F + F which becomes D – 2×F = 3×F Add 2×F to both sides of the above equation: D – 2×F + 2×F = 3×F + 2×F which makes D = 5×F
Hint #3
In eq.3, substitute 5×F for D: E = 5×F + F which makes E = 6×F
Hint #4
Substitute 6×F for E in eq.4: C = 6×F + F which makes C = 7×F
Hint #5
Substitute 2×F for B, and 7×F for C in eq.2: A = 2×F + 7×F which makes A = 9×F
Solution
Substitute 9×F for A, 2×F for B, 7×F for C, 5×F for D, and 6×F for E in eq.1: 9×F + 2×F + 7×F + 5×F + 6×F + F = 30 which simplifies to 30×F = 30 Divide both sides of the above equation by 30: 30×F ÷ 30 = 30 ÷ 30 which means F = 1 making A = 9×F = 9 × 1 = 9 B = 2×F = 2 × 1 = 2 C = 7×F = 7 × 1 = 7 D = 5×F = 5 × 1 = 5 E = 6×F = 6 × 1 = 6 and ABCDEF = 927561