Puzzle for July 30, 2020 ( )
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 B with C + E (from eq.3): A + C = C + E + E which becomes A + C = C + 2×E Subtract C from both sides of the above equation: A + C – C = C + 2×E – C which makes A = 2×E
Hint #2
In eq.2, replace A with 2×E: F = 2×E + C + E which becomes F = 3×E + C Subtract 3×E from each side of the above equation: F – 3×E = 3×E + C – 3×E which becomes eq.2a) F – 3×E = C
Hint #3
Substitute F – 3×E for C (from eq.2a) in eq.3: B = F – 3×E + E which becomes eq.3a) B = F – 2×E
Hint #4
In eq.4, substitute F – 3×E for C (from eq.2a), and 2×E for A: F – 3×E = 2×E + D Subtract 2×E from both sides of the equation above: F – 3×E – 2×E = 2×E + D – 2×E which becomes eq.4a) F – 5×E = D
Hint #5
Substitute F – 2×E for B (from eq.3a), and F – 5×E for D (from eq.4a) in eq.5: E + F = F – 2×E + F – 5×E which becomes E + F = 2×F – 7×E In the above equation, subtract F from both sides, and add 7×E to both sides: E + F – F + 7×E = 2×F – 7×E – F + 7×E which makes 8×E = F
Hint #6
Substitute 8×E for F in eq.3a: B = 8×E – 2×E which makes B = 6×E
Hint #7
Substitute 8×E for F in eq.2a: 8×E – 3×E = C which makes 5×E = C
Hint #8
Substitute 8×E for F in eq.4a: 8×E – 5×E = D which makes 3×E = D
Solution
Substitute 2×E for A, 6×E for B, 5×E for C, 3×E for D, and 8×E for F in eq.1: 2×E + 6×E + 5×E + 3×E + E + 8×E = 25 which simplifies to 25×E = 25 Divide both sides of the above equation by 25: 25×E ÷ 25 = 25 ÷ 25 which means E = 1 making A = 2×E = 2×E = 2 B = 6×E = 6×E = 6 C = 5×E = 5×E = 5 D = 3×E = 3×E = 3 F = 8×E = 8×E = 8 and ABCDEF = 265318