Puzzle for July 27, 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.5, replace D with A + F (from eq.4): C = A + F – A which makes C = F
Hint #2
In eq.6, replace F with C: B = C + C which makes eq.6a) B = 2×C
Hint #3
Add E to both sides of eq.3: B + E + E = D – E + E which becomes eq.3a) B + 2×E = D In eq.2, substitute B + 2×E for D (from eq.3a): B + 2×E + E = B + C which becomes B + 3×E = B + C Subtract B from both sides of the above equation: B + 3×E – B = B + C – B which makes 3×E = C and also makes F = C = 3×E
Hint #4
Substitute (3×E) for C in eq.6a: B = 2×(3×E) which makes B = 6×E
Hint #5
Substitute 6×E for B in eq.3a: 6×E + 2×E = D which makes 8×E = D
Hint #6
Substitute 3×E for F, and 8×E for D in eq.4: A + 3×E = 8×E Subtract 3×E from each side of the equation above: A + 3×E – 3×E = 8×E – 3×E which makes A = 5×E
Solution
Substitute 5×E for A, 6×E for B, 3×E for C and F, and 8×E for D in eq.1: 5×E + 6×E + 3×E + 8×E + E + 3×E = 26 which simplifies to 26×E = 26 Divide both sides of the above equation by 26: 26×E ÷ 26 = 26 ÷ 26 which means E = 1 making A = 5×E = 5 × 1 = 5 B = 6×E = 6 × 1 = 6 C = F = 3×E = 3 × 1 = 3 D = 8×E = 8 × 1 = 8 and ABCDEF = 563813